aa r X i v : . [ m a t h . F A ] F e b ON UNIFORMLY CONVEX FUNCTIONS
G. GRELIER, M. RAJA
Abstract.
Non-convex functions that yet satisfy a condition of uniform con-vexity for non-close points can arise in discrete constructions. We prove thatthis sort of discrete uniform convexity is inherited by the convex envelope,which is the key to obtain other remarkable properties such as the coercivity.Our techniques allow to retrieve Enflo’s uniformly convex renorming of super-reflexive Banach spaces as the regularization of a raw function built from trees.Among other applications, we provide a sharp estimation of the distance of agiven function to the set of differences of Lipschitz convex functions. Finally,we prove the equivalence of several natural fashions to quantify the non-superweakly compactness of a subset of a Banach space. Introduction
Along the paper, ( X, k · k ) will be a real Banach space and we will follow thestandard notation that one can find in books such as [2, 11, 13, 20, 23]. However,dealing with real functions defined on X , if there is not specific hypothesis onthe domain, we will follow the convention typical from Convex Analysis [4, 31]that a function f is defined everywhere and takes values in R = R ∪ {−∞ , + ∞} .A function f is said to be proper if f > −∞ and dom( f ) := { x ∈ X : f ( x ) < + ∞} 6 = ∅ . In the following, all the functions are supposed to be proper, howeversome operations performed on proper functions could lead to non proper functions.The class of lower semicontinuous convex proper functions on X will be denotedΓ( X ). Definition 1.1.
A function f defined on X is said to be ε -uniformly convex if fora given ε > , there is δ > such that if k x − y k ≥ ε then f (cid:18) x + y (cid:19) ≤ f ( x ) + f ( y )2 − δ. The function is said uniformly convex if it is ε -uniformly convex for all ε > . The suggestive name discrete uniformly convex functions to refer functionswhich are ε -uniformly convex for some ε > f is lower semicontinuous). The notion of uniform convexityin Analysis goes back to Clarkson [10], and uniform convexity for functions wasintroduced by Levitin and Polyak [22]. Since then, the properties of uniformly con-vex functions have been studied in several papers, notably [27, 28, 30, 1, 3, 5], thesection 3.5 in Zalinescu’s book [31] and part of chapter 5 in Borwein-Vaderwerff’s Date : February 4, 2021.The authors were supported by the Grants of Ministerio de Econom´ıa, Industria y Competi-tividad MTM2017-83262-C2-2-P; and Fundaci´on S´eneca Regi´on de Murcia 20906/PI/18. book [4] devoted to them. In relation to the standard theory, let us point out thenotion of modulus of uniform convexity δ f ( ε ) = inf (cid:26) f ( x ) + f ( y )2 − f (cid:18) x + y (cid:19) : x, y ∈ dom( f ) , k x − y k ≥ ε (cid:27) . Note that δ f could take negative values unless f is supposed to be (mid)convex.Analogously, it is possible to define ε -uniformly concave functions, however it willnot be necessary to treat them here because all the theory extends trivially.In this paper, we are focused in ε -uniformly convex functions for a fixed ε > regular convexity is not longer assumed . That is the main issuewe have to face deal here and the reason to do it is that non-convex ε -uniformlyconvex functions may arise in relation with some discrete constructions, startingfrom trees or barely convex sets. Nonetheless, ε -uniformly convex functions havenice properties. Along the paper, ˘ f will denote the lower semicontinous convexenvelope of function f (also denoted conv( f ) in some references). The next resultshows global behaviour of ε -uniformly convex functions and the relative stabilityof minimizers by linear perturbations. Theorem 1.2.
Let f be an ε -uniformly convex function such that ˘ f is proper.Then f is bounded below and coercive, more precisely we have lim inf k x k→ + ∞ f ( x ) k x k > . Moreover, for any ε ′ > ε there exists δ, η > such that if given x ∗ ∈ X ∗ and x ∈ X with f ( x ) + x ∗ ( x ) < inf( f + x ∗ ) + δ, and x ∗ ∈ X ∗ such that k x ∗ − x ∗ k < η and x ∈ X that minimizes f + x ∗ , then k x − x k ≤ ε ′ . The existence of such minimizer x is guaranteed if f = ˘ f . The proof of the former result relies in the possibility of “making convex” an ε -uniformly convex function without loosing the ε -uniformly convexity. We willsay that a function f is ε + -uniformly convex if it is ε ′ -uniformly convex for every ε ′ > ε . We have the following result. Theorem 1.3. If f is ε -uniformly convex, then ˘ f is ε + -uniformly convex. Simple examples, such as Example 2.6, shows that the ε -uniformly convexity of f does not guarantee that ˘ f would be proper. In order to have that dom( ˘ f ) = ∅ wemay ask f to be bounded below by an affine function (see Corollary 5.1). Supposethat we already have a proper lower semicontinuous convex and ε -uniformly convexfunction f . We wonder if we could “upgrade” f to a new function sharing thoseproperties and, besides, being locally Lipschitz (global Lipschitzness is not allowedfor uniformly convex functions). In that sense, we have the following result. Theorem 1.4.
Let f ∈ Γ( X ) be ε -uniformly convex function. Then there existsan equivalent norm ||| · ||| on X such that the function x → ||| x ||| is ε + -uniformlyconvex on bounded subsets of dom( f ) . Moreover, the norm ||| · ||| can be taken asclose to k · k as we wish. We want to point out that in the previous theorem we get ε + instead of ε de-spite the fact that the function f on the hypothesis is already convex. If f were N UNIFORMLY CONVEX FUNCTIONS 3 uniformly convex then a series of ε -uniformly convex norms for different ε ’s go-ing to 0 would produce an equivalent norm whose square is uniformly convex onbounded subsets of dom( f ).It turns out that supporting a convex continuous ε -uniformly convex functionis actually a geometrical-topological property of the domain. It is known thata Banach space admits a uniformly convex function bounded on bounded sets ifand only if it is super-reflexive. The second named author proved in [26] that aclosed convex bounded set admits a bounded continuous uniformly convex func-tion if and only if it is super weakly compact (SWC for short). We will give theactual definition of SWC set in Section 6, however we can provide an alternativeone on a provisional basis: a bounded closed convex set is SCW if and only iffor all ε > N ε such that the height of any ε -separated dyadic treesis bound by N ε . Recall that a dyadic tree of height n ∈ N is a set of the form { x s : | s | ≤ n } , indexed by finite sequences s ∈ S nk =0 { , } k of length | s | ≤ n , suchthat x s = 2 − ( x s⌢ + x s⌢ ) for every | s | < n , where { , } := {∅} indexes theroot x ∅ and the symbol “ ⌢ ” stands for concatenation. We say that a dyadic tree { x s : | s | ≤ n } is ε -separated if k x s⌢ − x s⌢ k ≥ ε for every | s | < n .Our techniques allow us to give a very precise quantitative version of the rela-tion between containment of separated trees and supporting a uniformly convexfunction for a set. Theorem 1.5.
Let C ⊂ X be a closed bounded convex set. Then these twonumbers coincide: ( a ) the infimum of the ε > such that there is a common bound for the heightsof all the ε -separated dyadic trees; ( b ) the infimum of the ε > such that there is a bounded ε -uniformly convex(and convex, Lipschitz. . . ) function defined on C . As we will see later, the quantities given by the previous theorem provide away to measure how far is a convex set of being non super weak compactness thatcompares to some many others. By the way, this result together Theorem 5.6 ap-plied to the ball of a Banach space produces a quantitative version of the famousEnflo’s renorming theorem of super-reflexive spaces. At this point, we want tostress that we are getting Enflo’s but not Pisier’s, see [11] for instance, because weare mainly focused on “ ε ” instead of “ δ ”. The quality of the modulus δ f will beworsened in many proofs being the only important thing that it remains positivefrom ε on. That is something that can be hardly done with other techniques tobuild convex functions like taking Minkowski functionals or distances of/from aconvex set.A couple of comments on the contents of this paper. We will consider the moregeneral notion of ε -uniformly convexity with respect to metric d , instead of thenorm. Namely, there is a metric d (uniformly continuous with respect to k · k by technical reasons) defined on dom( f ) such that there is δ > d ( x, y ) ≥ ε then f (cid:18) x + y (cid:19) ≤ f ( x ) + f ( y )2 − δ G. GRELIER, M. RAJA (the modulus δ f is defined likewise). With this definition Theorem 1.3, Theo-rem 1.4 and Theorem 1.5 are still true provided that dom( f ) is bounded. Itis known that the dual notion of uniform convexity is the uniform smoothness [1, 31, 3], however we will not discuss Fenchel duality here for ε -uniformly convexfunctions. That will be eventually done in a subsequent paper.The structure of the paper is the following. The second section deals withbasic properties of ε -uniformly convex and ε -uniformly quasi-convex functions,mostly under the hypothesis of convexity. A few examples are given to showthat the definitions do not guarantee some additional nice properties. The thirdsection is devoted to the proof of Theorem 1.3 that will allow the reduction tothe convex case of other results. The construction of uniformly convex functionsform scratch (trees and sets) is done in the fourth section. The fifth section treatsgeneral properties of ε -uniformly convex functions and the possibility of addingmore properties like Lipschitzness or homogeneity (renorming). We also provean estimation of the approximation by differences of convex functions. In thesixth section, we prove the equivalence of several measures of non super weakcompactness and the stability by convex hulls. In the last section we will sketchan understandable proof of Enflo’s uniformly convex renorming of super-reflexivespaces theorem based on our arguments.2. Basic properties and examples
We will discuss in this section results of almost arithmetical nature. The firstproposition contains some easy facts whose proof is left to the reader.
Proposition 2.1.
Let f be a proper ε -uniformly convex function. Then: (1) If g is convex, then f + g is ε -uniformly convex with δ f + g ≥ δ f . (2) The supremum of finitely many ε -convex functions is ε -convex too. (3) If f ≥ , then f is ε -uniformly convex. (4) The lower semicontinuous envelope of f is ε -uniformly convex. Recall that the infimal convolution of two functions f, g is defined as( f (cid:3) g )( x ) = inf { f ( x − y ) + g ( y ) : y ∈ X } . Proposition 2.2.
Let f , f be convex functions such that f is ε -uniformly con-vex and f is ε -uniformly convex. Then f (cid:3) f is ( ε + ε ) -uniformly convex withmodulus min { δ f ( ε ) , δ f ( ε ) } . Proof.
Given x , x ∈ dom( f (cid:3) f ) = dom( f )+dom( f ) with k x − x k ≥ ε + ε and η > y , y ∈ dom( f ) such that f ( x − y ) + f ( y ) < ( f (cid:3) f )( x ) + η,f ( x − y ) + f ( y ) < ( f (cid:3) f )( x ) + η. We have k ( x − y ) − ( x − y ) k + k y − y k ≥ k x − x k ≥ ε + ε . Therefore, one of the inequalities either k ( x − y ) − ( x − y ) k ≥ ε or k x − x k ≥ ε
2N UNIFORMLY CONVEX FUNCTIONS 5 holds. Assume the first one does (the other case is similar)( f (cid:3) f ) (cid:18) x + x (cid:19) ≤ f (cid:18) x + x − y + y (cid:19) + f (cid:18) y + y (cid:19) ≤ f ( x − y ) + f ( x − x )2 − δ f ε ) + f ( y ) + f ( y )2 ≤ ( f (cid:3) f )( x ) + ( f (cid:3) f )( x )2 − δ f ( ε ) + 2 η which implies the statement as η > Proposition 2.3.
Let f be convex and ε > . Then (1 − t ) f ( x ) + tf ( y ) − f ((1 − t ) x + ty ) ≥ δ f ( ε ) min { t, − t } whenever x, y ∈ dom( f ) , k x − y k ≥ ε and t ∈ [0 , . Proof.
Without loss of generality we may assume t ∈ [0 , /
2] so t = min { t, − t } .Note now that (1 − t ) x + ty = (1 − t ) x + 2 t x + y . By convexity of f we have f ((1 − t ) x + ty ) ≤ (1 − t ) f ( x ) + 2 tf (cid:18) x + y (cid:19) ≤ (1 − t ) f ( x ) + 2 t (cid:18) f ( x ) + f ( y )2 − δ f ( ε ) (cid:19) = (1 − t ) f ( x ) + tf ( y ) − tδ f ( ε )as wished.The gage of uniform convexity is introduced in [27] (see also [31, p. 203]) forconvex function as p f ( ε ) = inf (cid:26) (1 − t ) f ( x ) + tf ( y ) − f ((1 − t ) x + ty ) t (1 − t ) : 0 < t < , k x − y k ≥ ε (cid:27) . Corollary 2.4.
For any convex function f we have δ f ( ε ) ≤ p f ( ε ) ≤ δ f ( ε ) . Proof.
The first inequality is a consequence of the Proposition together with thefact that min { t, − t } ≥ t (1 − t ). The second inequality follows just taking t = 1 / ε -uniformly convexity can be expressedas p f ( ε ) >
0. The gage of uniform convexity has the following remarkable property p f ( λε ) ≥ λ p f ( ε )whenever ε ≥ λ ≥
1, see [31, Proposition 3.5.1] and note that the proofdoes not requiere the uniform convexity of f . In particular ε → ε − p f ( ε ) is a nondecreasing function.Now we will discuss some examples showing the limitations of the notions weare dealing with. G. GRELIER, M. RAJA
Example 2.5. f ( x ) = | x − / | is a continuous nonconvex -uniformly convexfunction on R . Proof.
This can be deduced by inspection of the drawing, and a more detailedcomputation shows that δ = 1 / ε -uniformlyconvex function to the set of convex functions does not depend on ε and can bearbitrarily large. Example 2.6.
A proper ε -uniformly convex function may have a non proper lowersemicontinuous convex envelope. Proof.
Take a function f which is finite and unbounded below on B (0 , ε/
2) andtakes as value + ∞ outside. By the very definition, f is ε -uniformly convex andnecessarily ˘ f = −∞ on B (0 , ε/ Example 2.7.
An unbounded uniformly convex continuous function taking finitevalues on a bounded convex closed set.
Proof.
Let C = B ℓ . It is obvious from the parallelogram equality that k x k isuniformly convex. Take a function h : [ − , → [0 , /
4] defined by h ( t ) = max { , t − / , − t − / } . The series g ( x ) = P ∞ n =1 nh ( x n ) for x = ( x n ) ∈ C defines a convex continuousfunction on C . Indeed, two summands cannot be positive at once on the samepoint. Since this is also true locally, the sum is continuous. The unbounded uni-formly convex continuous function is f ( x ) = k x k + g ( x ).The following notions will be useful in relation with ε -uniform convexity. Definition 2.8.
Let f : X → R be a function. Then f is said to be: (1) quasi-convex if f ( λx + (1 − λ ) y ) ≤ max { f ( x ) , f ( y ) } for every x, y ∈ X and λ ∈ [0 , . (2) ε -uniformly quasi-convex if there is some δ > such that f (cid:18) x + y (cid:19) ≤ max { f ( x ) , f ( y ) } − δ whenever x, y ∈ X with k x − y k ≥ ε (or d ( x, y ) ≥ ε for a metric). (3) uniformly quasi-convex if it is ε -uniformly quasi-convex for every ε > . Whereas the notion of quasi-convexity is well known, our definition of uniformquasi-convexity is weaker than the one given in [28]. As with convexity, the mid-point version does not implies the “ λ -version” unless some regularity (e.g. lowersemicontinuity) is assumed. The following result shows one relation between thequantified versions of uniform convexity and uniform quasi-convexity for functions. Proposition 2.9.
Let f ≥ be a convex ε -uniformly quasi-convex function. Then f is ε -uniformly convex. N UNIFORMLY CONVEX FUNCTIONS 7
Proof.
The following inequality can be checked easily: if for some real numbers a, b, c we have a + b ≥ c ≥ (cid:18) a + b − c (cid:19) + (cid:18) a − b (cid:19) ≤ a + b − c . Assume k x − y k ≥ ε and let δ > | f ( x ) − f ( y ) | > δ theprevious inequality implies f ( x ) + f ( y ) − f (cid:18) x + y (cid:19) ≥ δ . On the other hand, if | f ( x ) − f ( y ) | ≤ δ then f (cid:18) x + y (cid:19) ≤ max { f ( x ) , f ( y ) } − δ ≤ f ( x ) + f ( y )2 − δ f ( x ) + f ( y ) − f (cid:18) x + y (cid:19) ≥ δ Example 2.10. A ε -uniformly quasi-convex non-convex (concave) function. Proof.
Take f ( x ) = x for x < f ( x ) = x/ x ≥ Convexifying the ε -uniform convexity In order to cover previous developments around finite dentability [25] we willconsider uniformly convex functions with respect to a pseudometric d defined onthe domain of f . The norm of the Banach space will still play an importantrole and we requiere that d be uniformly continuous with respect to the norm.Therefore, along this section we will assume that ε -uniform convexity refers to d .We will refer as d -diameter of a subset in X × R the diameter with respect to d ofthe projection of the set onto X . Let ̟ be the modulus of uniform continuity (thestandard symbol is “ ω ” but we are using it as the first countable ordinal later),that is, the following inequality holds d ( x, y ) ≤ ̟ ( k x − y k )and lim t → + ̟ ( t ) = 0 . Proposition 3.1.
Let f : C → R be a function and ε > . Then (1) If f is ε -uniformly convex then every slice of epi( f ) disjoint from epi( f + δ f ( ε )) has d -diameter less than ε . (2) If f ∈ Γ( X ) and there is δ > such that every slice of epi( f ) disjoint from epi( f + δ ) has d -diameter less than ε then f is ε -uniformly convex withmodulus δ f ( ε ) ≥ δ/ . Proof.
For the first statement, assume that x, y belong to such a slice. Theseparation from epi( f + δ f ( ε )) implies f ( x ) + f ( y )2 < f (cid:18) x + y (cid:19) + δ f ( ε ) G. GRELIER, M. RAJA and so d ( x, y ) < ε . On the other hand, let δ > x, y ∈ X such that the following inequality holds f ( x ) + f ( y )2 − f (cid:18) x + y (cid:19) < δ . It implies that (cid:16) x + y , f ( x )+ f ( y )2 (cid:17) does not belong to epi( f + δ/ h such that h < f + δ/ h ( x + y ) > f ( x )+ f ( y )2 . It is evident thateither f ( x ) < h ( x ) or f ( y ) < h ( y ). We may assume without loss of generality thatthe first inequality holds as the scenario is symmetric for x and y . Now we have f ( y ) < h (cid:18) x + y (cid:19) − f ( x ) = h ( x ) + h ( y ) − f ( x ) < h ( y ) + δ . That implies both ( x, f ( x )) and ( y, f ( y )) belong to the slice defined by h + δ/ S = { ( x, t ) ∈ epi( f ) : t < h ( x ) + δ/ } . By our choices, we have S ∩ epi( f + δ ) = ∅ and thus d ( x, y ) < ε by the hypothesis.We deduce in this way that δ/ ≤ δ f ( ε ). Corollary 3.2.
Let f be a convex and ε -uniformly convex function. Then f ( x ) ≤ n X k =1 λ k f ( x k ) − δ f ( ε ) whenever x, x , . . . , x n ∈ dom( f ) satisfy that d ( x, x k ) ≥ ε and x = P nk =1 λ k x k with λ k ≥ and P nk =1 λ k = 1 . Proof.
If the inequality does not hold, then ( x, P nk =1 λ k f ( x k )) does not belongto epi( f + δ f ( ε )) so it can be separated from that set with a slice. Necessarily, oneof the points ( x k , f ( x k )) belongs to the slice. That implies that the d -diameter ofthe slice is at least ε which contradicts the previous proposition.The following result is based in the techniques of the geometrical study of theRadon-Nikodym property, see [6]. Lemma 3.3.
Let f : C → R be bounded below. Let m > be an upper bound forthe diameter of C and τ > and such that τ /m < . Assume that the set { x ∈ C : f ( x ) < inf f + δ } has diameter less than ε . Then the set { x ∈ C : ˘ f ( x ) < inf f + δτ /m } has diameter less than ε + 2 ̟ ( τ ) . Proof.
Consider the sets A = { ( x, r ) ∈ C × R : f ( x ) ≤ r < inf f + δ } ; B = { ( x, r ) ∈ C × R : r ≥ inf f + δ, f ( x ) ≤ r } . Note that the epigraph of f is A ∪ B . Consider their closed convex hulls ˘ A =conv( A ) and ˘ B = conv( B ) and note that conv( ˘ A ∪ ˘ B ) is dense in the epigraph of˘ f . Assume that ( x, r ) ∈ conv( ˘ A ∪ ˘ B ) and r < inf f + δτ /m . There is λ ∈ [0 , N UNIFORMLY CONVEX FUNCTIONS 9 such that ( x, r ) = λ ( y, t ) + (1 − λ )( z, s ) where ( y, t ) ∈ ˘ A and ( z, s ) ∈ ˘ B . Thecondition λt + (1 − λ ) s ≤ inf f + δτ /m implies 1 − λ < τ /m . Therefore k x − y k = k ( λ − y + (1 − λ ) z k = (1 − λ ) k y − z k < τ. In order to estimate the d -diameter of S = { ( x, t ) : x ∈ C, ˘ f ( x ) ≤ t < inf f + δτ /m } , we may consider only points on the dense set S ∩ conv( ˘ A ∪ ˘ B ). Therefore, consider( x , r ) , ( x , r ) ∈ conv( ˘ A ∪ ˘ B ) with r , r < inf f + δτ /m . The convex decompo-sition above shows that for some λ , λ ∈ [0 ,
1] and points ( y , t ) , ( y , t ) ∈ ˘ A and( z , s ) , ( z , s ) ∈ ˘ B we have( x , r ) = λ ( y , t ) + (1 − λ )( z , s ) , ( x , r ) = λ ( y , t ) + (1 − λ )( z , s ) . By the previous estimations, we have k x − y k , k x − y k ≤ τ which implies that d ( x , y ) , d ( x , y ) ≤ ̟ ( τ ), and thus d ( x , x ) ≤ d ( x , y ) + d ( y , y ) + d ( x , y ) ≤ ε + 2 ̟ ( τ )as desired. Theorem 3.4.
Let C ⊂ X be a closed convex bounded set and f : C → R bebounded below and ε -uniformly convex. Then ˘ f is ε + -uniformly convex and given ε ′ > ε , the modulus of convexity δ ˘ f ( ε ′ ) depends only on ε ′ , δ f ( ε ) , ̟ and thediameter of C . Proof.
Let m an upper bound for the diameter of C and δ > ε -uniform convexity. Take τ > τ /m < d -diameter of any slice of epi( ˘ f )not meeting epi( ˘ f + δτ /m ). Suppose that the slice is given by x ∗ ∈ X ∗ . Note thatthe estimation of the d -diameter of the slice we need is equivalent to the same foran horizontal slice of epi( ˘ f − x ∗ ) not meeting epi( ˘ f − x ∗ + δτ /m ), which is thesame that taking the points of epi( ˘ f − x ∗ ) whose scalar coordinate is less thaninf( ˘ f − x ∗ ) + δτ /m . Since ˘ f − x ∗ equals the convex envelope of the function f − x ∗ which is ε -uniformly convex with parameter δ , the set { x ∈ C : f ( x ) − x ∗ ( x ) < inf( f − x ∗ ) + δ } has diameter less than ε by Proposition 3.1. The previous lemma applies to getthat { x ∈ C : ˘ f ( x ) − x ∗ ( x ) < inf( f − x ∗ ) + δτ /m } has diameter less than ε + 2 ̟ ( τ ). Thanks to the Proposition 3.1, it follows that˘ f is ε + 2 ̟ ( τ )-uniformly convex. Given ε ′ > ε , we only have to set τ > ̟ ( τ ) < ε ′ − ε .The following result is the key to deal with unbounded domains. Proposition 3.5.
Let f be an ε -uniformly convex function such that ˘ f is proper.Then the value of ˘ f ( x ) for x ∈ dom( f ) depends only on the values of { f ( y ) : k y − x k < ε } . Namely, if g is the function defined by g ( y ) = f ( y ) if k y − x k < ε and g ( x ) = + ∞ otherwise, then ˘ f ( x ) = ˘ g ( x ) . Proof.
Note that the following set ( ( x, t ) : t ≥ n X k =1 λ k f ( x k ) with x = n X k =1 λ k x k a convex combination ) is dense in epi( ˘ f ). Fix x ∈ dom( f ) and suppose x = P nk =1 λ k x k is a convexcombination. We are going to describe an algorithm that will transform the set ofpoints S = { x , . . . , x n } into a set S ′ = { x ′ , . . . , x ′ n ′ } ⊂ B ( x, ε ) such that still wehave P n ′ k =1 λ ′ k x ′ k = x , where P n ′ k =1 λ ′ k x ′ k = x with λ ′ k ≥
0, and n ′ X k =1 λ ′ k f ( x ′ k ) ≤ n X k =1 λ k f ( x k ) . In order to do that, without loss of generality, we may assume x = 0. Fix x ∗ ∈ S X ∗ .Let a = sup { x ∗ , S } ≥ b = − inf { x ∗ , S } and suppose a ≥ max { b, ε } . Withoutloss of generality, we may assume x ∗ ( x ) = a . Since x is the further point (withrespect to x ∗ ), its “mass” λ compensates with masses on the side x ∗ ≤
0. Supposefirstly that x ∗ ( x ) ≤ λ ≥ λ . We have k x − x k ≥ ε . We claim that it ispossible to switch x by x ′ = ( x + x ) /
2. Indeed,2 λ x ′ + ( λ − λ ) x + λ x + · · · + λ n x n = 0which is still a convex combination. Note that2 λ f ( x ′ ) + ( λ − λ ) f ( x ) + λ f ( x ) + · · · + λ n f ( x n ) ≤ λ ( f ( x ) + f ( x )) + ( λ − λ ) f ( x ) + λ f ( x ) + · · · + λ n f ( x n )= λ f ( x ) + λ f ( x ) + λ f ( x ) + · · · + λ n f ( x n )where we have used f ( x ′ ) ≤ ( f ( x ) + f ( x )) /
2. The inequality means that S = { x ′ , x , . . . , x n } is an improvement of S in the sense of the approximation to ˘ f .Note also that x ∗ ( x ′ ) ≤ a/ λ > λ we will use several vectors x k with x ∗ ( x k ) ≤ x . This is possible because a ≥ b implies that the “mass” lying on the halfspace x ∗ < λ . In this case, λ could be cancelled with several λ k ’s.In any case, we will get a new set S whose cardinal is not larger than that of S and conv( S ) ⊂ conv( S ). After that, suppose that, unfortunately, we still havesup { x ∗ , S } = a . In such a case, the maximizing vector cannot be x , so it is anew vector, say x . We will apply the argument with x in order to replace it byanother vector x ′ and S by a new set S . Eventually, we will get sup { x ∗ , S n } < a after a finite number of steps. Then, with the same x ∗ , we have to change theconstants a, b > x ∗ until we getmax { a, b } < ε , so it is not possible to go further.If the set of points it is not yet inside B (0 , ε ) then find a new x ∗ ∈ S X ∗ suchthat sup { x ∗ , S n } ≥ ε and then run again the algorithm. Since conv( S ) is finitedimensional, it is enough to do this procedure over finitely many x ∗ ∈ S X ∗ inorder to get S n ⊂ B (0 , ε ) eventually. Remark 3.6.
An inspection of the proof of Proposition 3.5 shows that the algo-rithm used there could actually be pushed to obtain that ˘ f ( x ) depends on the valuesof f on a convex set of width less than ε around x . Note that the algorithm workswith a metric induced for a norm, not necessarily the main one. However, we donot know if it could work for a general metric. N UNIFORMLY CONVEX FUNCTIONS 11
Proof of Theorem 1.3.
If ˘ f is proper then it is bounded below by an affinefunction, so by adding an affine function (that does not alter the ε -uniform con-vexity), we may suppose that f is bounded below (actually that is true withoutmodifications, see Corollary 5.1). Given x, y ∈ dom( f ), if k x − y k ≥ ε then˘ f (cid:18) x + y (cid:19) ≤ ˘ f ( x ) + ˘ f ( y )2 − δ f ( ε ) . Indeed, fix η >
0. By Proposition 3.5, we may take ( x n ) a sequence of simple func-tions defined on [0 ,
1] (this is a convenient way to express a convex combination)such that k x n ( t ) − x k < ε for all t ∈ [0 , n ∈ N , withlim n Z x n ( t ) dt = x, and lim n Z f ( x n ( t )) dt = ˘ f ( x ) . Let ( y n ) and analogous sequence of simple functions playing the same role for y and ˘ f ( y ). Clearly we have k x n ( t ) − y n ( t ) k ≥ ε for all t ∈ [0 ,
1] and n ∈ N .Therefore ˘ f (cid:18) x + y (cid:19) ≤ lim inf n Z f (cid:18) x n ( t ) + y n ( t )2 (cid:19) dt ≤ lim n Z (cid:18) f ( x n ( t )) + f ( y n ( t ))2 − δ f ( ε ) (cid:19) dt ≤ ˘ f ( x ) + ˘ f ( y )2 − δ f ( ε ) . Since η > k x − y k ≥ ε .Now we will suppose ε ≤ k x − y k < ε . Proposition 3.5 implies that reducingthe domain of f to [ x, y ] + B (0 , ε ) does not affect to the values of ˘ f ( x ), ˘ f ( y ) and˘ f ( x + y ). Fix ε ′ > ε . Theorem 3.4 says that δ ˘ f ( ε ′ ) depends only on ε ′ , δ f ( ε ), ̟ ,which are fixed, and and the diameter of the domain, which bounded by 5 ε .4. Building uniformly convex functions
Most of the constructions of uniformly convex functions on a Banach spaces thatone can find in the literature are based on modifications of a uniformly convexnorm, see [5]. Nevertheless, the existence of a finite uniformly convex functionwhose domain has nonempty interior implies that X has an equivalent uniformlyconvex norm. In any case, the constructions dealing with the composition of auniformly convex norm and a suitably chosen function can be quite tricky, exceptfor the Hilbert space. Here we will exploit a method based on “discretization”and uniformly quasi-convex functions. Lemma 4.1.
Let f : C → R be a bounded below ε -uniformly quasi-convex withmodulus δ > . Then the function h ◦ f is ε -uniformly convex where h ( t ) = 3 t/δ . Proof.
The function h is increasing and satisfies the property 3 h ( t ) = h ( t − δ ).Take η = 4 − inf { h ( t + δ ) − h ( t ) : t ≥ inf f } = 2 − · inf f/δ and note that it depends only on f . If x, y ∈ C are such that d ( x, y ) ≥ ε take a = f ( x ), b = f ( y ) and c = f ( x + y ). The hypothesis says that c ≤ max { a, b } − δ .With no loss of generality, we may assume b ≤ a . We have h ( c ) ≤ h ( a ) − η. Since 3 h ( c ) ≤ h ( a ) and h ( b ) >
0, we also have3 h ( c ) ≤ h ( a ) + 2 h ( b ) and adding the previous inequality, we get4 h ( c ) ≤ h ( a ) + 2 h ( b ) − η and thus h ( c ) ≤ h ( a ) + h ( b )2 − η which is the ε -uniform convexity of h ◦ f .If X is uniformly convex, it is well known that x → k x k is a uniformly convexfunction on bounded convex subsets. The usual construction of a global uniformlyconvex functions involves additional properties of the norm, such as having a powertype modulus of uniform convexity. Here there is a simple alternative constructionbased in our methods. Proposition 4.2. If X has a uniformly convex norm then there exists a realfunction φ such that x → φ ( k x k ) is a uniformly convex function defined on X . Proof.
Fix ε >
0. Take a = ε/ a n ) by theimplicit equation a n − = (cid:18) − δ X (cid:18) εa n (cid:19)(cid:19) a n which has a unique solution thanks to the continuity of δ X on [0 , a n ) is increasing with lim n a n = + ∞ and has the followingproperty: if k x k , k y k ≤ a n and k x − y k > ε then k ( x + y ) / k ≤ a n − .Define a function as f ε ( x ) = n if a n − < k x k ≤ a n Note that f ε satisfies thehypothesis of Lemma 4.1 with δ = 1, and so h ◦ f ε is ε + -uniformly convex. Now,for ε = 1 /n take f n the convex envelope of h ◦ f ε and c n = 2 − n sup { f n , nB X } .The series P ∞ n =1 c n f n converges uniformly on bounded sets to a uniformly con-vex function. Clearly, the construction preserves the spheres centred at 0 as levelsets and therefore the function can be expressed through a composition with k · k .Now we will explain constructions using trees. The definition of ε -separated(dyadic) tree was given in the introduction. Bushes are defined in a very similarway, however the index set is S nk =0 N k and x s = P k λ s⌢k x s⌢k where λ s⌢k ≥ λ s⌢k = 0 except for finitely many k ’s and P k λ s⌢k = 1. We say that a bush { x s : | s | ≤ n } is ε -separated if k x s⌢k − x s k ≥ ε for all k such that λ s⌢k >
0. In thisway, an ε -separated tree is a particular case of a ε/ Proposition 4.3.
Let C be a convex set that supports a ε -uniformly convex func-tion with values in [ a, b ] . Then ( b − a ) /δ ( ε ) is the maximum height of (1) any ε -separated tree contained in C ; (2) any ε + -separated bush contained in C . Proof. If { x s } is an ε -separated tree then we have f ( x s ) ≤ max { f ( x s⌢ ) , f ( x s⌢ ) } − δ ( ε )that gives the estimation. In the case of bushes, the argument is the same afterpassing to ˘ f , which is ε + -uniformly convex by Theorem 1.3, and applying Corol-lary 3.2. N UNIFORMLY CONVEX FUNCTIONS 13
Our following result is quite a converse.
Theorem 4.4.
Let C be a convex set such that contains not arbitrarily high ε -separated trees (with respect some uniformly continous pseudometric). Then C supports a bounded ε -uniformly convex function and a bounded convex ε + -uniformly convex function (with respect the same pseudometric). Proof.
The function f ( x ) = − max { heigh( x s ) : ( x s ) ⊂ C ε -sep. tree , x ∅ = x } is ε -uniformly quasi-convex. Indeed, consider ε -separated trees of maximal heightwith roots x and y . Gluing both trees, after the eventual pruning of the tallerone, we will get a new ε -separated tree rooted at x + y . That means in terms ofthe function f the uniform quasi-convex inequality f (cid:18) x + y (cid:19) ≤ max { f ( x ) , f ( y ) } − d ( x, y ) ≥ ε . Now, Lemma 4.1 says that h ◦ f is ε -uniformly convex and itsconvex hull is the ε + -uniformly convex after Theorem 1.3. Proof of Theorem 1.5.
It just follows from Theorem 4.4 and Proposition 4.3.Finally we will explain constructions based on the dentability index. Let C bea bounded closed convex set of X , ( M, d ) a pseudometric space and F : C → M amap. We say that F is dentable if for any nonempty closed convex subset D ⊂ C and ε >
0, it is possible to find an open halfspace H intersecting D such thatdiam( F ( D ∩ H )) < ε , where the diameter is computed with respect to d . If F isdentable, we may consider the following “derivation”[ D ] ′ ε = { x ∈ D : diam( F ( D ∩ H )) > ε, ∀ H ∈ H , x ∈ H } . Here H denotes the set of all the open halfspaces of X . Clearly, [ D ] ′ ε is whatremains of D after removing all the slices of diameter less or equal than ε . Auseful trick is the so called (nonlinear) Lancien’s midpoint argument: if a segmentsatisfies [ x, y ] ⊂ D and [ x, y ] ∩ [ D ] ′ ε = ∅ then d ( F ( x ) , F ( y )) ≤ ε , see the beginningof [25, Theorem 2.2]. Consider the sequence of sets defined by [ C ] ε = C and, forevery n ∈ N , inductively by [ C ] nε = [[ C ] n − ε ] ′ ε . If there is n ∈ N such that [ C ] n − ε = ∅ and [ C ] nε = ∅ we say that Dz( F, ε ) = n .We say that F is finitely dentable if Dz( F, ε ) < ω for every ε > ω stands forthe first infinite ordinal number). All these notions can be applied to the identitymap of a convex set where there is a pseudometric defined. The following resultis the quantified version of [25, Theorem 2.2]. For convenience we will write∆ Φ ( x, y ) = Φ( x ) + Φ( y )2 − Φ (cid:18) x + y (cid:19) . Theorem 4.5.
Let F : C → M be a uniformly continuous map from a convex setto a pseudometric space and ε > . (1) Suppose that there exists a bounded lower semi-continuous convex function Φ defined on C and δ > such that d ( F ( x ) , F ( y )) ≤ ε whenever x, y ∈ C satisfy ∆ Φ ( x, y ) ≤ δ , then Dz(
F, ε ) < ω . (2) On the other hand, if
Dz(
F, ε ) < ω then for every ε ′ > ε there exits abounded lower semi-continuous convex function Φ defined on C and δ > such that d ( F ( x ) , F ( y )) ≤ ε ′ whenever x, y ∈ C satisfy ∆ Φ ( x, y ) ≤ δ . Proof.
Let s = sup { f, C } . The hypothesis implies [ C ] ′ ε ⊂ { f ≤ s − δ } . Iterat-ing this we will eventually get to the emptyset. For the second part, we need tointroduce some notation. Firstly put d ′ ( x, y ) = d ( F ( x ) , F ( y )) which is a pseudo-metric uniformly continuous with respect to k · k . Derivations and diameters willbe referred to d ′ . The slice of a set A with parameters x ∗ ∈ X ∗ and α > S ( A, x ∗ , α ) = { x ∈ A : x ∗ ( x ) > sup { x ∗ , A } − α } . The “half-derivation” of a convex set is defined as h D i ′ ε = { x ∈ D : x ∗ ( x ) ≤ α, ∀ x ∗ , α > s.t. diam( S ( D, x ∗ , α )) > ε } . The geometric interpretation is that we remove half of the slice, in sense of thewidth, for every slice of d ′ -diameter less than ε . This derivation can be iteratedby taking h C i nε = hh C i n − ε i ′ ε . It is not difficult, but rather tedious, to show thatif Dz( F, ε ) < ω then for every ε ′ > ε there is some N ∈ N such that h C i nε ′ = ∅ .The idea is the following. Firstly note that every slice of C not meeting [ C ] ′ ε hasdiameter 2 ε at most by Lancien’s argument. Taking “half a slice” of the slice givenby some x ∗ ∈ X ∗ , we deduce thatsup { x ∗ , h C i ′ ε } − sup { x ∗ , [ C ] ′ ε } ≤ − (sup { x ∗ , C } − sup { x ∗ , [ C ] ′ ε } ) . Iterating, we would getsup { x ∗ , h C i n ε } − sup { x ∗ , [ C ] ′ ε } ≤ − n (sup { x ∗ , C } − sup { x ∗ , [ C ] ′ ε } )for every x ∗ ∈ X ∗ . If η >
0, we will get for some n large enough that h C i n ε ⊂ [ C ] ′ ε + B (0 , η ) . We can do that for every set [ C ] kε . A perturbation argument, using the roombetween ε and ε ′ , will allow us to fill the gap between the sequences of sets. Inthis way we will get that h C i nε ′ = ∅ for some n ∈ N large enough.Now we define a function g on C by g ( x ) = − n if x ∈ h C i nε ′ \ h C i n +1 ε ′ following thenotation above. We claim that g satisfies Lemma 4.1 with separation ε ′ . Indeed,if d ′ ( x, y ) > ε ′ and n = − max { g ( x ) , g ( y ) } then x, y ∈ h C i nε ′ . If x + y
6∈ h C i n +1 ε ′ then the segment [ x, y ] would be fully contained into a slice of diameter less than ε ′ and so d ′ ( x, y ) ≤ ε ′ which is a contradiction. Therefore x + y ∈ h C i n +1 ε ′ and so g ( x + y ) ≤ − n −
1. Now f ( x ) = 3 g ( x ) is ε ′ -uniformly convex with respect to d ′ .Take Φ = ˘ f to get the desired function.If F in Theorem 4.5 (2) were finitely dentable, a standard argument using aconvergent series would lead to this results, which is essentially [15, Proposition3.2 ] with a uniformly convex function instead of a norm. Corollary 4.6.
Let F : C → M be a uniformly continuous finitely dentable map.Then there exits a bounded convex function Φ defined on C such that for every ε > there is δ > such that d ( F ( x ) , F ( y )) ≤ ε whenever x, y ∈ C are such that ∆ Φ ( x, y ) ≤ δ . N UNIFORMLY CONVEX FUNCTIONS 15 Improving functions and domains
So far the best improvement we have done on an existing ε -uniformly convexfunction is taking its lower semicontinuous convex envelope provided this last oneis proper. The aim in this section is to manipulate the functions in order to im-prove their qualities. We will begin by proving the results about global behaviour. Proof of Theorem 1.2 . Since ˘ f ≤ f it is enough to prove the property holds fora ε -convex and convex proper function. Actually the same proof for a uniformlyconvex function done in Zalinescu’s book [31, Proposition 3.5.8] works in this casebecause lim inf t → + ∞ t − p f ( t ) ≥ ε − p f ( ε ) > x ∗ = 0 (justtake change f by f + x ∗ ). Take δ = δ ˘ f ( ε ′ ) and take η = inf f + δ − ˘ f ( x ) > f = inf ˘ f . By the property established in the first part applied to˘ f − x ∗ , there is R > f ( x ) ≥ ˘ f ( x ) − x ∗ ( x − x ) for any x ∗ ∈ B X ∗ and k x − x k ≥ R . Now, fix x ∗ such that k x ∗ k ≤ η/R . Then we have˘ f ( x ) + x ∗ ( x ) ≥ ˘ f ( x ) + x ∗ ( x ) − δ for all x ∈ X such that k x − x k ≤ R , and therefore the inequality holds for all x ∈ X . That implies epi( ˘ f + x ∗ + δ ) does not meet the horizontal slice S = { ( x, t ) ∈ epi( ˘ f + x ∗ ) : t ≤ ˘ f ( x ) + x ∗ ( x ) } By Proposition 3.1 S has diameter less than ε ′ . Moreover, if f + x ∗ attains aminimum at x , then the same holds for ˘ f + x ∗ and so x ∈ S . Since x ∈ S wehave k x − x k ≤ ε ′ . The existence of a dense set of x ∗ such that ˘ f + x ∗ attains aminimum is guaranteed by Brøndsted-Rockafellar [4, Theorem 4.3.2] (or Bishop-Phelps [13, Theorem 7.4.1] applied to the epigraph).As a consequence, we characterize when an ε -uniformly convex function has aproper convex envelope. Corollary 5.1.
Let f be a ε -uniformly convex function not identically + ∞ . Thenthe following statements are equivalent: (1) ˘ f is proper; (2) f is bounded below; (3) f is bounded below by an affine continuous function. For a ε -uniformly quasi-convex function we can say the following Proposition 5.2.
Let f be a ε -uniformly quasi-convex function that is boundedbelow. Then f is coercive and moreover lim inf k x k→ + ∞ f ( x ) k x k > . Proof.
By adding a constant, we may suppose that inf f = 0. Take x ∈ X such that f ( x ) < δ/
2. For any x ∈ X such that k x − x k ≥ ε we have f ( x ) ≥ δ . Indeed, otherwise it would be f ( x ) < δ and by ε -uniformly quasi-convexity f ( x + x ) < inf f , an obvious contradiction. Now, if k x − x k ≥ ε , then k x + x − x k ≥ ε . That implies f ( x + x ) ≥ δ and therefore f ( x ) ≥ δ . Inductively,we will get that if k x − x k ≥ n ε then f ( x ) ≥ n δ . Now, the statement follows easily.The following results will show that, given a ε -uniformly convex function, wecan made modification in both the function and its domain in order to get a newfunction with additional properties. Proposition 5.3.
Let f be a ε -uniformly convex function that is locally boundedbelow and η > . Then there exists a lower semicontinuous ( ε + 2 η ) -uniformlyconvex function defined on dom( f ) + B (0 , η ) . In particular, dom( f ) admits alower semicontinuous ε + -uniformly convex function. Proof.
Define g ( x ) = inf { f ( y ) : k y − x k < η } on dom( f ) + B (0 , η ). This function g is ( ε + 2 η )-uniformly convex (the simple verification of this fact is left to thereader). Now take its lower semicontinuous envelope.The following result will be done for ε -uniformly convexity with respect to ametric because such a degree of generality will be needed later. Proposition 5.4.
Let f be an ε -uniformly convex function (with respect to ametric d with modulus of uniform continuity ̟ ) and let C ⊂ dom( f ) be convexsuch that f is bounded there. Then for any ε ′ > ε , there exists g ∈ Γ( X ) Lipschitz(with respect to the norm of X ) such that g | C is ε ′ -uniformly convex. Proof.
Without loss of generality, we may assume that f is convex. Take η > ̟ ( η ) < ( ε ′ − ε ) / m = sup { f ( x ) − f ( y ) : x, y ∈ C } and c = m/η . Define g ( x ) = inf { f ( y ) + c k x − y k : y ∈ C } which is convex and c -Lipschitz. Suppose x ∈ C . Then either g ( x ) = f ( x ) andthe infimum is attained with y = x , or g ( x ) < f ( x ). In the last case, the infimumcan be computed over the y ∈ C such that f ( y ) + c k x − y k < f ( x ). Therefore, k x − y k < m/c = η . In any case, to each x ∈ C we can assign y ∈ C with d ( x, y ) < ( ε ′ − ε ) / x , x ∈ C with d ( x , x ) ≥ ε ′ find y , y ∈ C approximating the infimum lessthan some ξ >
0. Clearly we have d ( y , y ) ≥ ε , and so g (cid:18) x + x (cid:19) ≤ f (cid:18) y + y (cid:19) + c (cid:13)(cid:13)(cid:13)(cid:13) x + x − y + y (cid:13)(cid:13)(cid:13)(cid:13) ≤ f ( y ) + f ( y )2 − δ + c k x − y k + c k x − y k ≤ g ( x ) + g ( x )2 − δ + 2 ξ. As ξ > ε ′ -uniform convexity of g as wished. Remark 5.5.
A Baire category argument shows that an ε -uniformy convex func-tion f is bounded in an open ball if dom( f ) has nonempty interior. However wedo not know how to ensure that f will be bounded on a larger set. Now we will show how to change an ε -uniformly convex function by a normwith the same property. Theorem 5.6.
Let f ∈ Γ( X ) be a non negative function and C ⊂ dom( f ) abounded convex set. Assume f is Lipschitz on C . Then given δ > there existsan equivalent norm ||| · ||| on X and ζ > such that ∆ f ( x, y ) < δ whenever x, y ∈ C satisfy ∆ |||·||| ( x, y ) < ζ . As a consequence, if f was moreover ε -uniformly convex(with respect to a pseudometric) on C , then ||| · ||| would be ε -uniformly convex on C . N UNIFORMLY CONVEX FUNCTIONS 17
Proof.
Taking f ( x ) + f ( − x ) + k x k instead, we may indeed assume that f issymmetric and attains a strong minimum at 0. Let M = sup { f, C } and m =min { f, C } + δ/
2. The Lipschitz condition easily implies that there is η > r ≤ M then { f ≤ r − δ } + B (0 , η ) ⊂ { f ≤ r } . For r ∈ [ m, M ] let k · k r the Minkowski functional of the set { f ≤ r } which is anequivalent norm on X . Let N = sup {k x k : x ∈ C } and note that λ = (1 + η/N ) − has the property { f ≤ r − δ } ⊂ λ { f ≤ r } . We deduce the following property: if x, y ∈ C , k x k r , k y k r ≤ f ( x, y ) ≥ δ then (cid:13)(cid:13)(cid:13)(cid:13) x + y (cid:13)(cid:13)(cid:13)(cid:13) r ≤ λ. Consider a partition m = a < a < · · · < a k = M such that a j /a j +1 < λ / andput ||| · ||| j = k · k a j . Let x, y ∈ C such that ∆ f ( x, y ) ≥ δ . Assume f ( x ) ≥ f ( y ) forinstance. There is some 1 ≤ j < k such that 1 ≥ ||| x ||| j ≥ λ / . Since ||| x + y ||| j ≤ λ ,we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ≤ max {||| x ||| j , ||| y ||| j } − ( λ / − λ ) . Following the same computations that in the proof of Proposition 2.9, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ≤ ||| x ||| j + ||| y ||| j − ( λ / − λ ) . Therefore, if we define an equivalent norm by ||| · ||| = P kj =1 ||| · ||| j we will have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ||| x ||| + ||| y ||| − ( λ / − λ ) . whenever x, y ∈ C satisfies ∆ f ( x, y ) ≥ δ , meaning that the statement is true with ζ = 4 − ( λ / − λ ) . Proof of Theorem 1.4.
By Proposition 5.4, we may assume that f is alreadynorm Lipschitz and finite on X provided we change ε by ε + . Theorem 1.2 impliesthat f is bounded below and any bounded subset C ⊂ dom( f ) will be eventually“captured” by a level set { f ≤ n } with n ∈ N . Let ||| · ||| n the norm given byTheorem 5.6 on the bounded set { f ≤ n } which is ε + -uniformly convex there.Take α > α n ) a sequence of positive numbers such that ||| · ||| = α ||| · ||| + ∞ X n =1 α n ||| · ||| n converges uniformly on bounded sets. Clearly, ||| · ||| will be ε + -uniformly convexon bounded sets too. The last affirmation follows just taking α > D C -Lipschitz if it is thedifference of two convex Lipschitz functions. The symbol k · k C stands for thesupremum norm on the set C . Theorem 5.7.
Let C ⊂ X be a bounded closed convex set and let f : C → R bea uniformly continuous function. Consider the following numbers: ( ε ) the infimum of the ε > such that Dz( f, ε ) < ω ; ( ε ) the infimum of the ε > such that there exists a D C -Lipschitz function g such that k f − g k C < ε .Then ε / ≤ ε ≤ ε . Proof.
Let ε > ε and find a D C -Lipschitz function g such that k f − g k C < ε .We know by [25, Proposition 5.1] that g is finitely dentable, which easily impliesthat f is 2 ε -finitely dentable.For the reverse inequality, take ε > ε and M = sup { f ( x ) − f ( y ) : x, y ∈ C } < + ∞ . Apply Theorem 4.5 to get a function Φ such that | f ( x ) − f ( y ) | ≤ ε if ∆ Φ ( x, y ) < δ . By Proposition 5.4 we may suppose that Φ is Lipschitz too,and by Theorem 5.6, there is an equivalent norm ||| · ||| defined on X such that∆ |||·||| ( x, y ) < ζ implies ∆ Φ ( x, y ) < δ . Take c > M/ζ . Consider g = inf y ∈ C ( f ( y ) + c ||| x ||| + ||| y ||| − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) !) = inf y ∈ C { f ( y ) + c ∆ |||·||| ( x, y ) } . For every x ∈ C , the infimum can be computed just on the set A ( x ) = { y ∈ C : f ( y ) + c ∆ |||·||| ( x, y ) } ≤ f ( x ) } If x ∈ C and y ∈ A ( x ), we have0 ≤ c ∆ |||·||| ( x, y ) ≤ f ( x ) − f ( y ) ≤ M. Then ∆ |||·||| ( x, y ) ≤ ζ by choice of c and thus 0 ≤ f ( x ) − f ( y ) ≤ ε . Fix η > y ∈ A ( x ) such that f ( y ) + c ∆ |||·||| ( x, y ) ≤ g ( x ) + η, then f ( x ) − g ( x ) ≤ f ( x ) − f ( y ) − c ∆ |||·||| ( x, y ) + η ≤ ε + η We deduce that k f ( x ) − g ( x ) k C ≤ ε and g ( x ) = c ||| x ||| − sup y ∈ C ( c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − c ||| y ||| − f ( y ) ) which is an explicit decomposition of g as a difference of two convex Lipschitzfunctions on C , as wanted.6. Quantifying the super weakly compactness
The notion of super weak compactness was introduced in [9], however someresults were established independently in [25] for an equivalent notion (finitedentable) within the convex sets. Here we will use a definition based on ultra-powers. Given a free ultrafilter U on a set N , recall that X U is the quotient of ℓ ∞ ( X ) by the subspace of those ( x i ) i ∈ N such that lim i, U k x i k = 0. We take K U as the image of K by the canonical embedding x ( x ) i ∈ N . A subset K ⊂ X issaid to be relatively super weakly compact if K U is a relatively weakly compactsubset of X U for a (equivalently all) free ultrafilter U on N , and K is said to be super weakly compact if it is moreover weakly closed. The following result gathersseveral equivalent properties, see [9, 25, 26, 21], in order to compare with theirquantified versions (Theorem 6.3). N UNIFORMLY CONVEX FUNCTIONS 19
Theorem 6.1.
Let C ⊂ X be a bounded closed convex subset. The followingstatements are equivalent: (1) given ε > it is not possible to find arbitrarily long sequences x , . . . , x n ∈ C such that d (conv { x , . . . , x k } , conv { x k +1 , . . . , x n } ) ≥ ε for all k = 1 , . . . , n − ; (2) C contains not arbitrarily high ε -separated dyadic trees for every ε > ; (3) C U is relatively weakly compact in X U for U a free ultrafilter on N (equiv-alently, C is relatively SWC, by definition); (4) C U is dentable in X U for U a free ultrafilter on N ; (5) C is finitely dentable; (6) C supports a convex bounded uniformly convex function. In order to state our results we need to introduce some quantities related tosets in Banach spaces. Firstly, a measure of non weakly compactness that hasbeen studied in several papers [14, 17, 7], see also [18, Section 3.6]. Given A ⊂ X consider A ⊂ X ∗∗ by the natural embedding and take γ ( A ) = inf { ε > A w ∗ ⊂ X + εB X ∗∗ } . It turns out that γ ( A ) = 0 if and only if A is relatively weakly compact, thus γ quantifies the non weakly compact of subsets in X . This measure is consideredmore suitable than De Blasi’s measure for some problems in Banach space theory.Given a convex set A ⊂ X , let us denote by Dent( A ) the infimum of the numbers ε > A has nonempty slices contained in balls of radius less than ε ,and take ∆( A ) = sup { Dent( C ) : C ⊂ A } . The measure Dent was introduced in[8] in relation with the quantification of the RNP property, and actually we maythink of ∆ as a measure of non RNP. Lemma 6.2.
Let A ⊂ X be a closed convex bounded subset, U a free ultrafilteron N and ε > . Then [ A U ] ′ ε ⊂ ([ A ] ′ ε ) U . Proof.
Given ( x n ) ∈ A U \ ([ A ] ′ ε ) U , we have to find a slice of A U containing ( x n )of diameter not greater than 2 ε . As ( x n ) ([ A ] ′ ε ) U , then for some α > { n : d ( x n , [ A ] ′ ε ) ≥ α } ∈ U . Indeed, otherwise the sequence ( x n ) would be equivalent to a sequence in [ A ] ′ ε . Itis possible to find x n ∈ B X ∗ such that x ∗ n ( x n ) ≥ α + sup { x ∗ n , [ A ] ′ ε } for those indices n from the previous set, for the other n ’s the choice of x n ∈ B X ∗ does not makea difference. Consider the functional ( x ∗ n ) ∈ ( X ∗ ) U ⊂ ( X U ) ∗ . By construction, h ( x ∗ n ) , ( x n ) i ≥ α + sup { ( x ∗ n ) , ([ A ] ′ ε ) U } . Now, we will estimate the diameter of theslice defined by ( x ∗ n ). Suppose that( y n ) , ( z n ) ∈ A U ∩ { ( x ∗ n ) ≥ α + sup { ( x ∗ n ) , ([ A ] ′ ε ) U }} . Then for a subset in U of indices n , we have y n , z n ∈ A ∩ { x ∗ n ≥ α + sup { x ∗ n , A n }} and thus k y n − z n k ≤ ε by Lancien’s midpoint argument. That implies k ( y n ) − ( z n ) k ≤ ε , so the diameter of the slice is not greater than ε as wished.The following result is the quantitative counterpart of Theorem 6.1. Theorem 6.3.
Let C ⊂ X be a bounded closed convex subset. Consider thefollowing numbers: ( µ ) the supremum of the numbers ε > such that for any n ∈ N there are x , . . . , x n ∈ C such that d (conv { x , . . . , x k } , conv { x k +1 , . . . , x n } ) ≥ ε forall k = 1 , . . . , n − ; ( µ ) the supremum of the ε > such that there are ε -separated dyadic trees ofarbitrary height; ( µ ) = ∆( C U ) for U a free ultrafilter on N ; ( µ ) = γ ( C U ) for U a free ultrafilter on N ; ( µ ) the infimum of the ε > such that Dz(
C, ε ) < ω ; ( µ ) the infimum of the ε > such that C supports a convex bounded ε -uniformly convex function.Then µ ≤ µ ≤ µ ≤ µ ≤ µ and µ ≤ µ ≤ µ ≤ µ . Proof.
We will label the steps of the proof by the couple of numbers associatedto the inequality.(1-2) If ε < µ , the separation between convex hulls applied to 2 n elements allowsthe construction of a ε -separated dyadic trees of height n . Therefore µ ≥ µ .(2-3) If ε < M then ∆( C U ) ≥ ε/
2. Indeed, C U contains an infinite ε -separateddyadic tree T , therefore any nonempty slice of T cannot be covered by finitelymany balls of radius less than ε/ A ) ≤ γ ( A ), therefore ∆( C U ) ≤ γ ( C U ).(4-1) Let ε < µ . Then there is x ∈ C w ∗ which is at distance greater than ε from X . Following Oja’s proof of James theorem [13, Theorem 3.132], it is posibleto find an infinite sequence ( x n ) with convex separation greater than ε . Finiterepresentativity gives arbitrarily large sequences in X with the same separation,thus µ ≥ µ .(4-5) If ε > M then there is a finite sequence of sets C = C ⊃ C ⊃ · · · ⊃ C n given by the ε -dentability process. Taking weak ∗ closures in the bidual, we have C w ∗ = ( C w ∗ \ C w ∗ ) ∪ · · · ∪ ( C n − w ∗ \ C nw ∗ ) ∪ C nw ∗ . Now, take any x ∈ C w ∗ belongs to one of those sets. The w ∗ -open slice separating x from the smaller set, say C k +1 w ∗ ( ∅ for the last set) in the difference is containedin the w ∗ -closure of a slice of C k not meeting C k +1 which has diameter less than 2 ε (Lancien’s midpoint argument). Since w ∗ -closures does not increase the diameter,we have d ( x, X ) ≤ ε . The argument actually implies γ ( C ) ≤ ε , however we canapply it to the sequence of sets in X U C U = C U ⊃ C U ⊃ · · · ⊃ C U n which has the same slice-separation property by Lemma 6.2.(5-6) If ε > µ , there is a bounded convex ε -uniformly convex function f that,without loss of generality, we may suppose lower semicontinuous. By Proposition3.1, any slice of the set { x ∈ C : f ( x ) ≤ a } not meeting the set { x ∈ C : f ( x ) ≤ a + δ } has diameter less than ε . A judicious arranging of these sets shows that C is ε -finitely dentable. Thus µ ≤ µ .(6-2) Take ε > µ . Then the ε -separated dyadic trees are uniformly bounded inheight. By Theorem 4.4, that implies the existence of ε ′ -uniformly convex functionfor every ε ′ > ε . Thus µ ≤ µ . N UNIFORMLY CONVEX FUNCTIONS 21
Remark 6.4.
The equivalence between µ and µ is both a local and a quantitativeversion of the well know statement saying that super-RNP is the same that super-reflexivity. Let us point out that some other relations between the quantities µ i for i = 1 , . . . , can be established and so improving the equivalence constants. Forinstance µ ≤ µ which is somehow straightforward or µ ≤ µ as a consequenceof Proposition 4.3. We will need the following estimation of how much thicken the closure withrespect to the topology induced by a norming subspace of the dual.
Lemma 6.5.
Let X a Banach space and F ⊂ X ∗ an -norming subspace. Thenfor any bounded convex A ⊂ X and any ε > γ ( A ) we have A σ ( X,F ) ⊂ A + 2 εB X . Proof.
By [18, Proposition 3.59], A w ∗ ⊂ A + 2 εB X ∗∗ . The linear map p : X ∗∗ → F ∗ defined by p ( x ∗∗ ) = x ∗∗ | F has norm 1 and satisfies p ( A w ∗ ) = A σ ( F ∗ ,F ) . We mayidentify p ( X ) = X isometrically into F ∗ and so we have A σ ( F ∗ ,F ) ⊂ A + 2 εB F ∗ .Therefore A σ ( X,F ) ⊂ A + 2 εB X as wished.We will need the following result that appears as a fact inside the proof of [29,Theorem 3.1]. The 1-norming subspace ( X ∗ ) U ⊂ ( X U ) ∗ will play an importantrole. Lemma 6.6.
For any ( x ∗ n ) ∈ ( X ∗ ) U and ( a n ) ∈ conv( A ) U , there is ( b n ) ∈ conv( A U ) such that h ( x ∗ n ) , ( a n ) i ≤ h ( x ∗ n ) , ( b n ) i . Among the quantities given by Theorem 6.3 only µ does not requiere convexity,so we can propose it as a natural measure of non super weak compactness. Thefollowing is a quantitative version (in terms of µ ) of [29, Theorem 3.1] establishingthat the super weak compactness is stable by closed convex hulls. Note that themeasure of non super weak compactness introduced by K.Tu in [29] is differentfrom ours and so our result is not equivalent to [29, Theorem 4.2]. Theorem 6.7.
Let A ⊂ X be a bounded subset and U a free ultrafilter. Then γ (conv( A ) U ) ≤ γ ( A U ) . Proof.
Consider F = ( X ∗ ) U which is an 1-norming subspace of ( X U ) ∗ . Take ε > γ ( A U ). By Lemma 6.5,conv σ ( X,F ) ( A U ) ⊂ conv( A U ) + 2 εB X U . We claim (conv( A )) U ⊂ conv( A U ) + 2 εB X U . If it not the case, then we couldseparate a point (conv( A )) U from conv σ ( X,F ) ( A U ) by a functional from F . Thatleads to a contradiction with Lemma 6.6. Now, we have γ ((conv( A )) U ) ≤ γ ( A U ) + 2 ε which implies the statement.7. A new glance at Enflo’s theorem
Let us show how Enflo’s theorem follows from our results.
Theorem 7.1 (Enflo [12]) . Let X be a super-reflexive Banach space. Then X hasan equivalent uniformly convex norm. Proof.
The unit ball B X endowed with the weak topology is SWC. Therefore,there is a bounded convex ε -uniformly convex function defined on B X for every ε > k · k ε on X whose square is an ε -uniformly convex function on B X . Without loss of gener-ality, we may assume that k · k ≤ k · k ε ≤ k · k . The series ||| · ||| = P ∞ n =1 − n k · k /n defines an equivalent uniformly convex norm.Enflo’s original proof of the uniformly convex renorming of super-reflexive Ba-nach spaces has remain practically unchanged in books, see [13, Pages 438-442]for instance. We believe that the reason is that the proof is difficult to follow froma geometrical point of view. One of the original aims of this paper was to castsome light on the renorming of super-reflexive spaces. Since the geometrical ideasare now diluted along this paper, we would like to offer to the interested reader amore direct pathway to Enflo’s theorem as a successive improvement of functions. • From the usual definition of super-reflexivity with finite representation, itis easy to prove that the unit ball B X of a super-reflexive space has thefinite tree property, that is, given ε > ε -separated dyadic trees [19]. • The maximal heigh of an ε -separated tree with root x ∈ B X is an ε -uniformly concave function h ( x ). This is the main idea in the proof ofTheorem 4.4. Note that this function is also symmetric. • g ( x ) = 3 − h ( x ) is a symmetric ε -uniformly convex function taking values in[0 , • f = ˘ g is convex, symmetric and 3 ε -uniformly convex. The key idea is that f ( x ) is computed with the values of g ( y ) with k y − x k < ε . The technicaldetails can be carried out as in the proof of Theorem 1.3, which relies onProposition 3.5, nevertheless the idea is very intuitive: Planet Earth is anon-convex ε -uniformly convex radial body for ε = 800 km at most. Thatimplies you do not need the Rocky Mountains neither the Himalayas tocompute the convex hull over France. • Let f n the function given in the previous steep for ε = 1 /n . The function F ( x ) = k x k + ∞ X n =1 − n f n ( x )is uniformly convex, symmetric, Lipschitz on the balls rB X for 0 < r < F (0) ≤ /
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