Support, Convexity Conditions and Convex Hypersurfaces in Infinite Dimension
aa r X i v : . [ m a t h . F A ] S e p Support, Convexity Conditions and ConvexHypersurfaces in Infinite Dimension
Paolo d’AlessandroFormer Professor at the Dept of Math Third Univ. of [email protected] 17, 2019
Abstract
Working in infinite dimensional linear spaces, we deal with supportfor closed sets without interior. We generalize the Convexity Theorem forclosed sets without interior. Finally we study the infinite dimensional ver-sion of Jordan hypersurfaces. Our whole work never assumes smoothnessand is based exclusively on non-differential Convex Analysis tools and, inparticular, on theory of convex cones. A crucial mathematical tool for ourresults is obtained solving the decomposition problem for non-closed nonpointed cones.AMS Classification: 52A07; 53D99; 53B10
If we consider a convex set C in a real linear topological space E and assumethat it has interior, then, by a well known topological separation principle, eachpoint in B ( C ) is a support point for C .It is also well known that, conversely, if each point of the boundary of abody C has support, then C is convex. These two results form what we call theConvexity Theorem for Bodies (i.e. sets with interior).The hypothesis of the Theorem is considered of global nature in the litera-ture. And a main stream of literature is that of deducing the same conclusionfrom purely local conditions. Usually smoothness is assumed, the local hy-pothesis is provided by curvature and, naturally, the machinery of differentialgeometry takes over. Since the present paper does not follow this stream ofresearch we just cite [6] as an example.Here we move with a different orientation. First and foremost we never as-sume smoothness of surfaces and stay clear of manifold theory. In this non-smooth setting we lean instead only on non-differential Convex Analysis tech-niques and find our main toolset in the theory of convex cones. This approachwill allow us to give support results for closed sets without interior, general-ize the Convexity Theorem to closed sets without interior, and, in a Hilbert1pace setting, develop a non-linear range space theory of convex hypersurfaces.Moreover, in the final Section, we introduce and investigate the concept of con-vexification of a hypersurface.We believe that the basic concept of support should not be considered global,simply because it is non-differential. Indeed it is a local non-smooth condition,a fact which we try to substantiate within our theory.The conical techniques we use are geared on polarity, and, in our non-smoothsetting, the concept of tangent cone is the simpler possible. We have no use herefor its differential version, namely the famous Bouligant cone.Interestingly, within the present vision of geometry of convex sets, linear andnon-linear range space theory present some superpositions. Moreover, workingon these topics a connection between topology and convexity, typical of ConvexAnalysis, does persists. In this Section we make precise some technicalities of Vector Topology, thatmake possible starting a formal treatment.When we consider a linear topological space (lts) or locally convex space(lcs), we will assume for simplicity that the space be real. It is instead forsubstantial reasons, appearing in a short while in connection to boundedness,that we will soon restrict ourselves to Hausdorff lts.A terminological warning. As in [1] we call sphere what in geometry is calledball, and what in geometry is called sphere is for us the boundary of a ball.Also in this paper all cones are intended to be convex. That is, C is a conein a lts E if C + C ⊂ C and αC ⊂ C for any real α ≥ Proposition 1
Let E be a lts, then { } − = ∩{ U : U is a neighborhood of } Moreover, { } − is always a linear subspace. The space E is Hausdorff if andonly if: { } − = { } (A explicit proof is given for example in [7] at the end of p.29) Thus if a spaceis not Hausdorff then { } − is a non-trivial linear subspace so that { } − = { } ). Incidentally, what one could (and should) do when the space E is not Haus-dorff is to consider the topological quotient space E/ { } − , which is a Hausdorfflts and is in fact the right Hausdorff representation of E . Even more inciden-tally, this is the correct formalization of the a.e. concept in functions spaces.An in-depth analysis might be advisable though, because this is not an issueto be dismissed as lightly as usual. One for example wants to verify that the2inear subspace { } − stays fixed across the various topologies considered for agiven space. But even more serious consideration deserve the facts stated byTheorem 8 in above reference.In the present context this facts are relevant because of the following con-sequence of the equation { } − = { } , characterizing Hausdorff spaces, whichregards boundedness.Notice in this respect that boundedness is translation independent. Alsowe call ray (emanating from the origin) the conical extension (or hull) of anon-zero vector. Corollary 2
Consider a Hausdorff lts E . Then a necessary condition for a set C in E to be bounded is that it does not contain any ray or any translated ray. Proof.
Without restrictions of generality we may assume that 0 ∈ C . Supposethat C is bounded but contains a ray. Then because C is absorbed by everyneighborhood of the origin, it follows that all neighborhoods must contain thesame ray. But then the intersection of all such neighborhoods contains such aray and this contradicts the hypothesis that the space be Hausdorff, becausethe equation { } − = { } cannot hold.Naturally, we will make in what follows use of many elementary computa-tions on convexity, see Theorem 13.1 in [1]. We recall a few statement, that willbe of use here. Let E be a lts and A a subset of E , then: A convex ⇒ A i , A − convex A convex body ⇒ tA i + (1 − t ) A − ⊂ A i , 0 < t ≤ A convex body ⇒ A i = A ♦ ; A i − = A − ; A − i = A i where A ♦ denotes the radial kernel of A . And for any set B : C − ( B ) = C ( B ) − where is the convex extension (or hull) of B and C − ( B ) is the closed convexextension (or hull) of B .We will make of course intensive use of Separation Principles. We recall twostatements from [1], so that we can bear in mind the exact hypotheses of eachmajor separation result. Theorem 3
Suppose E is a lts and A and B (non-void) convex sets. Suppose A has interior and that B ∩ A i = φ . Then there exists a linear continuousfunctional that separates A and B . Notice that E is not supposed to be Hausdorff in this particular result.Second Theorem . This time we speak of locally convex spaces (again notnecessarily Hausdorff): Theorem 4
Suppose E is a lcs and A and B (non-void) disjoint convex sets,with A compact and B closed. Then there exists f ∈ E ∗ that strongly separates A and B . The Basic Convexity Theorem
We consider a convex body C (that is, a convex set with interior) in a realHausdorff lts E , and prove the following Theorem, which gives a little additionalinformation with respect to the Convexity Theorem cited at the beginning. Inthis respect it is useful to bear in mind that whereas any point of B ( C ) is asupport point of C in view of the first Separation Theorem, it is instead obviousthat no point of C i can be a support point, because each such point has aneighborhood entirely contained in C i . Theorem 5
Consider a closed body C in a lts. Then C is convex if and onlyif each of its boundary points is a support point for C , and, if this is the case,then C is the intersection of any family of closed semispaces obtained choosinga single support closed semispace for each point of B ( C ) itself. Proof.
The only if is immediate consequence of the first cited separation The-orem. Then we prove that if each of the boundary points of C is a supportpoint for C itself, then C is the intersection specified in the statement, andconsequently also a closed convex body. Without restriction of generality andsince translation is a homeomorphism that leaves invariant all convexity prop-erties, we can assume that 0 ∈ C i . Consider the intersection Ψ specified in thestatement. Then Ψ is a closed convex set containing C and thus a closed convexbody too. Suppose there is a point y in Ψ such that y / ∈ C or y ∈ C . Since C is an open set, there is a neighborhood U of y contained in C itself. Also thereis a neighborhood W of the origin with W ⊂ C i Consider the ray ρ generatedby y. By radiality of neighborhoods there is a segment [0 : z ] ⊂ W with z = 0and [0 : z ] ⊂ ρ . We take v = βy with β = sup { γ : γy ∈ C i } . This sup existsbecause y / ∈ C i and because C i is convex and so no point γy with γ > C i . Also β > z . Moreover, v cannot be in C i because otherwise it would have a neighborhood contained in C i and since neighborhood of v are radial at v we would contradict that β is thegiven sup. But then it must be v ∈ B ( C ), since there is a net (in ρ ∩ C i ) thatconverges to v by construction, and therefore by the first Separation Theoremit is a support point for C . At this point if v = y we are done, because wehave found a contradiction. If not it must be β <
1. Consider the separatingcontinuous linear functional f corresponding to v . We have f (0) = 0, andto fix the ideas and without restriction of generality, f ( v ) = βf ( y ) >
0. Butthen, because f ( y ) > f ( v ), the supporting functional leaves the origin and y on opposite open semispaces. This would imply that y / ∈ Ψ, which is again acontradiction. Thus the proof is completed.Of course all the more C is the intersection of all the closed supportingsemispaces of C . Remark 6
Notice that this Convexity Theorem is an instance of the loose con-nection with topology often met in Convex Analysis. In fact if a closed set hasno interior we may well try to strenghten the topology of the ambient spaceto the effect of forcing the interior to appear. Another example of this loose onnection, but going the other way around, is the celebrated Krein -MilmanTheorem, where we may weaken the topology in an effort to induce compactnessof a convex set, to the effect of making the KM Theorem applicable. We Start making more explicit from the preceding proof an important propertyof convex bodies.
Theorem 7
Suppose C is a closed convex body in a lts and that, without re-striction of generality, ∈ C i . Then C = ∪{ l ∩ C , l is a ray } and, if a ray l meets B ( C ) , the intersection is a singleton. Therefore if { z } issuch an intersection [0 : z ) ⊂ C i . Proof.
The first part is obvious, since if y ∈ C then [0 : y ] ⊂ C . As to thesecond statement suppose that z, w ∈ ρ ∩ B ( C ) and assume for example that z is closer to the origin than w. Then, since C has support at z , the supportinghyperplane will leave the origin and w on opposite open semispaces. This is acontradiction and so the second part is proved too. Finally the last part is adirect consequence of elementary computations on convex sets.With reference to the above Theorem, we prove, regarding boundedness andunboundedness the following Theorem 8
Suppose C is a closed convex body in a Hausdorff lts and that,without restriction of generality, ∈ C i . If there exists a ray, that does notmeet B ( C ) , then C is unbounded. Thus if C is bounded all the rays meet B ( C ) (in a unique point). Proof.
Consider a ray ρ which does not meet B ( C ) and suppose that thereexists a vector z ∈ ρ ∩ C . Because C is open ρ ∩ C is open in ρ . Consider β = inf { α : αz ∈ ρ ∩ C } Notice that β > C has interior. It cannot be w = βz ∈ ρ ∩ C , becausethere would be a neighborhood of w contained in C and, by radiality of neigh-borhoods in a lts, we would contradict the definition of β . Therefore w ∈ C and, more precisely, w ∈ B ( C ) since each of its neighborhoods meets C . Thiscontradicts that the ray does not meet B ( C ), and therefore such a vector z cannot exist. Thus, if a ray ρ does not meet B ( C ), then ρ ⊂ C . But then byCorollary 2, C is unbounded. This completes the proof.Theorem 7 has a Corollary which, in a way, generalizes the finite dimensionalfact asserting that if C is compact C − ( C ) = C ( C ). This peculiar fact for finitedimension can be proved leaning on the celebrated Caratheodory’s Theorem,which is finite dimensional only as well.5 orollary 9 Suppose C is a closed bounded convex body in a Hausdorff lts andthat, without restriction of generality, ∈ C i . Let y ( ρ ) be the unique pointwhere each ray ρ meets B ( C ) . Then C = ∪{ [0 : y ( ρ )] }B ( C ) = { y ( ρ ) : ρ is a ray } C = ∪{ [ x : y ] : x, y ∈ B ( C ) } and C − ( B ( C )) = C ( B ( C )) = C Proof.
The first three statements are either obvious consequences of the pre-ceding Theorem or of immediate proof. As to the last statement, to simplifynotations, let Ψ = ∪{ [ x : y ] : x, y ∈ B ( C ) } and notice that: C ⊃ C − ( B ( C )) ⊃ C ( B ( C )) ⊃ Ψ = C and so we are done.This result allow us to rephrase the Convexity Theorem for bounded bodiesin an interesting way. Theorem 10
Consider a closed bounded body C in a Hausdorff lts. Then C is convex if and only each point of B ( C ) supports B ( C ) itself, and if this is thecase, then C is the intersection of any family of semispaces obtained choosing asingle closed semispace supporting B ( C ) for each point of B ( C ) . Proof. the proof follows from the fact that C = C ( B ( C )), and thus a closedsupporting semispace contains C if and only if it contains B ( C ).Again a fortiori C is the intersection of all semispaces that support B ( C ).That a set (not necessarily convex and possibly without interior) has supportat a point in its boundary is equivalent to the fact that the normal cone at suchpoint is not trivial (meaning that such a cone is = { } ). The normal cone isnothing but the polar of the tangent cone. We recall a few basic concepts andproperties first about cones and then about polarity. Definition 11
Let V be a cone in a lts E . The lineality space of V is the linearsubspace lin ( V ) : lin ( V ) = V ∩ ( − V ) Clearly lin ( V ) is the maximal subspace contained in V . We say that V is pointedif lin ( V ) = { } . Beware that Phelps ([3]) calls proper cone what is for us a pointed cone.Here instead a proper cone is a cone whose closure is a proper subset of theambient lts.
Definition 12
A cone in a lts E is proper if it is not dense in the whole space.Thus a closed cone is proper if it is not the whole space, or, equivalently, is aproper subset of E .
6e next cite from [5] a first major result regarding support for void interiorconvex sets, illustrating the mathematical facts and reasoning from which itoriginates. These recalls will be instrumental in the sequel.
Lemma 13
A cone in a lcs E is proper (equivalently its closure is a propersubset) if and only if it is contained in a closed half-space. Proof.
Sufficiency is obvious. As to necessity, let C be a closed proper cone.Then there is a singleton { y } disjoint from C . Singletons are convex and com-pact and therefore the second Separation Theorem cited at the beginning ap-plies. Thus there exists a continuous linear functional f such that: f ( x ) < f ( y ), ∀ x ∈ C and since C is a cone this implies: f ( x ) ≤ ∀ x ∈ C Thus the condition is also necessary and we are done.
Theorem 14
The closure of a pointed cone in a linear topological space is aproper cone.
Proof.
Suppose that it is not true, that is there is a pointed cone C in alinear topological space E , such that C − = E . Consider a finite dimensionalsubspace F , with its (unique) relative topology, which intersect C in a non triv-ial, necessarily pointed, cone. Actually we can take instead of F , its subspace L ( F ∩ C ), without restriction of generality. For simplicity we leave the symbol F unchanged. Next notice that, as is well known, because F is the finite dimen-sional, the pointed convex cone Υ = F ∩ C has interior. Thus it can be separatedby a continuous linear functional from the origin and therefore it is containedin a closed semi-space. It follows that the closure of Υ in F is contained in aclosed half-space and therefore is a proper cone. But by Theorem 1.16 in [8],such closure is C − ∩ F . By the initial assumption C − ∩ F = E ∩ F = F . Thisis a contradiction and therefore the proof is finished. Definition 15
Let C be a closed subset of a lts E and x ∈ B ( C ) . Then theTangent cone T C ( x ) to C at x is the cone: T C ( x ) = C o ( C − x )We now turn to lcs spaces, because we start using duality for linear topo-logical spaces. Definition 16
Let C be a subset of a lcs E . Then the polar of C is the (alwaysclosed) convex cone: C p = { f : f ∈ E ∗ , f ( x ) ≤ , ∀ x ∈ C } efinition 17 Let C be a closed subset of a lcs E and x ∈ B ( C ) . Then theNormal cone N C ( x ) to C at x is the cone: N C ( x ) = T C ( x ) p Also we will use the other important notion of Normal Fan. Let sp ( C ) theset of all points at which a closed set C has support. Definition 18
Let C be a closed subset of a lcs E . Then the Normal Fan N F ( C ) is the set: N F ( C ) = ∪{ N C ( x ) : x ∈ sp ( C ) } The following Theorem has an immediate proof and nevertheless in conjunc-tion with the above Lemma 13 and Theorem 14 it allows to state a major resultconcerning support for sets with void interior.
Theorem 19
Consider a closed set C in a lcs. Then a point y ∈ B ( C ) isextreme if and only if the cone T ( y ) is pointed. Proof.
Straightforward consequence of the definition of extreme point.Let’s gather for locally convex spaces some important consequences of theanalysis developed in this Section. The fact that the closure of a pointed coneis a proper cone (Theorem 14) and Lemma 13 (valid for locally convex spaces)imply that the polar cone of a pointed cone is always non-trivial. At this pointwe can apply Theorem 19 to conclude that for a closed set (without interior) if apoint in its boundary is extreme, then the tangent cone at that point is pointedand hence its polar, or normal cone at the point in question, is non-trivial. Westate this result for sets without interior, since if there were a non-void interior,support would be insured anyway. We summarize the above discussion in thefollowing
Theorem 20
Consider a closed set without interior in a lcs. Then it has sup-port at all points of its boundary that are extreme.
This is a first fundamental result on support for convex closed sets with voidinterior. It has been applied by the author to the Theory of Maximum Principleas outlined in the following
Remark 21
This result (Theorem 20) was stated in [5], where it was used ina Hilbert Space setting to demonstrate that the Maximum Principle holds for allcandidate targets. We briefly sketch the meaning of this assertion. Considera linear PDE in a Hilbert space setting and suppose that a target ζ is reach-able from the origin in an interval [0 , T ] . Then that a minimum norm control u o (steering the origin to ζ in the time interval [0 , T ] ) exists is an immediateconsequence of the fact that the forcing operator L T is continuous and of theProjection Theorem. Let ρ be the norm of u o . The Maximum Principle holdsif ζ is a support point for L T ( S ρ ) (where S ρ is the closed sphere of radius ρ inthe L space of input functions), which is a closed convex set without interior n general. Now it is shown, in the cited paper, that ζ is always a support pointfor L T ( S ρ ) because ζ is always extreme, and hence Theorem 20 applies. Thefact that ζ is extreme is proved exploiting a property that all Hilbert space have,namely strict convexity. For details proofs and much more the interested readeris referred to the cited paper We saw that the closure of a pointed cone is proper. What if the cone is notpointed? We will investigate this issue in the real Hilbert Space environment (inthe sequel all Hilbert spaces are assumed to be real without further notice), andthen, after developing the appropriate mathematical tools, we will introduce acompletely general necessary and sufficient condition of support for closed setwith void interior.Naturally in a Hilbert space environment we can take advantage of the RieszTheorem and so we can consider Normal Cones as subsets of the Hilbert spaceitself.
Definition 22
Let C be a closed subset of a lcs E . Then the Applied NormalFan AN F ( C ) is the set: AN F ( C ) = ∪{ x + N C ( x ) : x ∈ sp ( C ) } To begin with we recall some relevant material from [2].We start from an elementary Lemma.
Lemma 23
Suppose that F and G are closed subspaces of Hilbert space H ,that F ⊥ G , and that, for two non-void subsets C and D of H , C ⊂ F and D ⊂ G . Then C + D is closed if and only if both C and D are closed. Moreover: y = P F y + P G y ∈ C + D ⇐⇒ P F y ∈ C and P G y ∈ D and if C and/ or D are not closed ( C + D ) − = C − + D − Proof.
We can assume without restriction of generality that G = F ⊥ because,if this were not the case we can take in lieu of H the Hilbert space H = F + G ,which is in fact a closed subspace of H . Suppose that C and D be closed andconsider a sequence { z i } in C + D with { z i } → z . We can write in a uniqueway: z i = P F z i + P G z i with P F z i ∈ C and P G z i ∈ D . By continuity of projection and the assumptionthat C and D be closed , { P F z i } → P F z ∈ C and { P G z i } → P G z ∈ D , butsince P F z + P G z = z it follows z ∈ C + D and we are done. Conversely suppose9hat, for example D is not closed so that there exists a sequence { d i } in D , thatconverges to d / ∈ D , but, of course, d ∈ G . Take a vector c ∈ C . The sequence { c + d i } converges to c + d . But c = P F ( c + d ) and d = P G ( c + d ), and therefore,by uniqueness of the decomposition, it is not possible to express c + d as a sumof a vector in C plus a vector in D . It follows that c + d / ∈ C + D . The secondstatement follows immediately from uniqueness of decomposition of a vector y in the sum P F y + P G y , and since the proof of the last statement is immediate,we are done.Next it is in order to recall from the same source the fundamental theoremon decomposition of non-pointed cones. We give a slightly expand the statementand include the proof for subsequent reference. Theorem 24
Consider a convex cone C in a Hilbert space H and assume thatits lineality space be closed. Then ( lin ( C ) ⊥ ∩ C ) = P lin ( C ) ⊥ C where the cone lin ( C ) ⊥ ∩ C is pointed. Consequently, if C is closed the cone P lin ( C ) ⊥ C is closed too. Moreover, the cone C can be expressed as: C = lin ( C ) + ( lin ( C ) ⊥ ∩ C ) = lin ( C ) + P lin ( C ) ⊥ C Finally if a cone C has the form C = F + V , where F is a closed subspace and V a pointed cone contained in F ⊥ , then F is its lineality space and the givenexpression coincides with the above decomposition of the cone. Proof.
First we prove that C = lin ( C ) + ( lin ( C ) ⊥ ∩ C )That the rhs is contained in the lhs is obvious. Consider any vector x ∈ C andfor brevity let Γ = lin ( C ). Decompose x as follows: x = x Γ + x Γ ⊥ (1)where x Γ ∈ Γ and x Γ ⊥ ∈ Γ ⊥ . Next x Γ ⊥ = x − x Γ as sum of two vectors in C is in C and hence in Γ ⊥ ∩ C . Thus we have proved that the lhs is containedin the rhs . Next we show that the cone lin ( C ) ⊥ ∩ C is pointed. Suppose thatboth a vector x = 0 and its opposite − x belong to lin ( C ) ⊥ ∩ C and decompose x as above (1). Because lin ( C ) ⊥ ∩ C ⊂ lin ( C ) ⊥ , x Γ = 0, so that x = x Γ ⊥ = 0.Do the same for − x , to conclude that x Γ ⊥ and − x Γ ⊥ are in C (but obviouslynot in lin ( C )). Because this is a contradiction, lin ( C ) ⊥ ∩ C is pointed. Finallywe prove that: lin ( C ) ⊥ ∩ C = P lin ( C ) ⊥ C In fact, P Γ ⊥ (Γ ⊥ ∩ C ) = Γ ⊥ ∩ C ⊂ P Γ ⊥ ( C )10n the other hand if z ∈ P Γ ⊥ ( C ), for some w ∈ C , z = P Γ ⊥ w = w − P Γ w so that z ∈ C . Hence z ∈ Γ ⊥ ∩ C and so P Γ ⊥ ( C ) ⊂ Γ ⊥ ∩ C . Finally, let x ∈ C = F + V as defined in the last statement. Write x = x F + x F ⊥ . According to Lemma23 x F ∈ F and x F ⊥ ∈ V . Now − x = − x F − x F ⊥ and, by the same Lemma, − x ∈ C if and only if − x F ∈ F and − x F ⊥ ∈ V . But V is assumed to be pointedand so − x F ⊥ = 0, which shows that F = lin ( F + V ). On the other hand thepointed cone of the decomposition is P F ⊥ ( F + V ) = V and thus the proof iscomplete.In general, and in particular for tangent cones, we cannot make any closed-ness assumption. Thus we need to develop a mathematical tool, which allow usto deal with the case non closed cones. We will tackle this problem momentarily,right after taking a short break to dwell on polarity.Polarity can be viewed as the counterpart of orthogonality in the context ofcone theory, as already seen from the following basic computations for polars.We do not pursue this parallel in depth here, limiting ourselves to what is neededhere (for more details see [2]). Many of the following formulas are for genericsets, however we will be interested mostly in the special cases of polars of cones. Proposition 25
The following formulas regarding polars hold where C and D are arbitrary sets ( − C ) p = − C p ( C p ) p = C − C − p = C p C ⊂ D ⇒ D p ⊂ C p ( C + D ) p = C p ∩ D p Moreover, if F is a linear subspace (closed or non closed it doesn’t matter) then F p = F ⊥ Remark 26
The polar cone of a convex cone is the normal cone at the originto the given convex cone. Also, the polar cone of a closed convex cone is theset of all points in the space, whose projection on the cone, coincides with theorigin.
We still have to handle the case where the lineality space of a cone is notclosed, and here comes our solution for this issue.
Remark 27
In the next Theorem we exclude the case that the cone C is con-tained in lin ( C ) , for in this case since the reverse inclusion always holds wehave C = linC that is, the cone is a linear subspace. Such case is trivial sincewe have only to ask, for our purposes, that the cone is not dense. heorem 28 Consider a non pointed non closed cone C and let Γ be its lin-eality space, which is assumed to be not dense. We also assume that C is notcontained in Γ and that Γ is not closed. Consider the set Ψ = (( C \ Γ − ) ∪ { } ) ∩ Γ ⊥ ⊂ C Then Ψ is a non-void pointed cone, and if ∆=Γ − + Ψ then lin ∆=Γ − (so that and Theorem 24 applies to ∆ ) and: C ⊂ ∆ ⊂ C − Moreover, C and ∆ have the same polar cone. Proof.
We first show that Ψ is a cone. Referring to non-zero vectors to avoidtrivialities, first of all it is obvious that x ∈ Ψ implies αx ∈ Ψ ∀ α ≥
0. If x, y ∈ Ψ then z = x + y ∈ Γ ⊥ and z ∈ C , and these two imply that z ∈ Ψ.Thus Ψ is a cone. At this point we also know that ∆ is a cone, because it is thesum of two cones. Next if there where two non-zero opposite vectors in Ψ theywould also be in C \ Γ (since C \ Γ − ⊂ C \ Γ) contradicting that Γ is the linealityspace of C . Therefore Ψ is pointed. Now we look at the intersection:(Γ − + Ψ) ∩ ( − Γ − − Ψ) = (Γ − + Ψ) ∩ (Γ − − Ψ)with the aid of Lemma 23. Clearly, for a vector with a component in Ψ, to bein the intersection, it is required that we find two opposite vectors in Ψ, whichis impossible because Ψ is pointed. Thus a non-zero vector in the intersectioncan only be in Γ − and so we have proved that lin ∆=Γ − . As the first inclusionin the statement (namely C ⊂ ∆), suppose x = 0 is in C . Either x ∈ Γ − andthus x ∈ ∆ or x ∈ C \ Γ − . In this latter case we write x = x Γ − + x Γ ⊥ . Nowarguing as in the proof of Theorem 24, P Γ ⊥ ( C ) ⊂ Γ ⊥ ∩ C (by the way this alsoshows that Ψ is non-void) and so x Γ ⊥ ∈ Γ ⊥ ∩ C . Thus it must be x Γ ⊥ ∈ Ψ.So we have proved that in any case x ∈ ∆ and so we have proved that C ⊂ ∆.Next, applying Lemma 23 to ∆, we get:∆ − = Γ − + Ψ − But Γ ⊂ C ⇒ Γ − ⊂ C − and, similarly, Ψ ⊂ C ⇒ Ψ − ⊂ C − . Thus ∆ − is thesum of two cones, both contained in C − , and therefore we got ∆ − ⊂ C − and,a fortiori, ∆ ⊂ C − . This completes the proof of the statement about inclusions.Now applying the elementary computations on polars we have: C − p ⊂ ∆ p ⊂ C p C − p = C p it follows: C p = ∆ p Thus the proof is finished.Sometimes, when a cone is contained in a subspace, it is convenient to con-sider the polar of a cone within the subspace itself, regarded as the ambientspace. When we do so we put a subscript, indicating the subspace, under thesymbol of the polar.We now establish another important result on polar cones.
Theorem 29
Suppose that a cone C has the form F + Ψ , where F is a closedproper subspace and Ψ is a pointed cone in F ⊥ . Then C p = Ψ pF ⊥ Therefore the polar of the cone is contained in F ⊥ and in view of Theorem 14cannot be trivial. Proof.
By direct computation. First of all (by elementary computations onpolars) C p = F ⊥ ∩ Ψ p , which implies C p ⊂ F ⊥ . Thus if x ∈ C p then x = x F ⊥ .Therefore for y ∈ C , the inequality ( x, y ) ≤ x ∈ Ψ pF ⊥ , as we wantedto prove.Applying the last two Theorems, we reach the conclusion contained in thestatement of the following Theorem, which obviously requires no proof: Theorem 30
Consider a non pointed non closed cone C and let Γ be its lin-eality space, which is assumed to be not dense. We also assume that C is notcontained in Γ and that Γ is not closed. Then by Theorem 28 it is true that: C p = ∆ p where ∆=Γ − + Ψ and Ψ is the pointed cone contained in Γ ⊥ defined in the statement of 28. Thus C p = Ψ pF ⊥ in view of Theorem 29 The mathematical tools that we have been developing so far, put together,allow us to state the following fundamental Theorem on support for closed setswith void interior in Hilbert spaces.
Theorem 31
Consider a closed set C with void interior in a Hilbert space H .Then a point y ∈ B ( C ) has support if and only if the tangent cone T C ( y ) iseither pointed (that is, y is an extreme point) or is not pointed but its linealityspace is not dense. Proof.
The proof is contained in the theory developed so far and does notrequire any additional effort. 13
Generalizing the Convexity Theorem to Setswithout Interior
We recalled at the beginning the Convexity Theorem asserting that a closedbody is convex if and only if each point of its boundary has support.Here, sticking to the Hilbert space environment we remove the conditionthat the set has interior. Throughout the rest of the paper the ambient spaceis always understood to be a real infinite dimensional Hilbert space (with theonly exception of our very last Theorem).We start with a Lemma.
Lemma 32
Consider a closed set C without interior (so that C = B ( C ) ) andsuppose x ∈ C has support. Then for all the points of x + N C ( x ) the minimumdistance problem from both C and C − ( C ) has the unique solution given by x . Proof.
By definition of normal cone, for any the point z in x + N C ( x ) theminimum distance problem of z from x + T C ( x ) − has a unique solution givenby x . Let δ = k z − x k . Now, since C ⊂ T C ( x ) − , k w − x k > δ , ∀ w ∈ C .Therefore x , which is in C , is also the unique minimum distance solution of z from C . Since C − ( C ) ⊂ x + T C ( x ) − the same argument shows that x , which isalso in C − ( C ), is the unique minimum distance solution of z from C − ( C ) . Thiscompletes the proof .
Corollary 33
Consider a closed set C without interior (so that C = B ( C ) ) andsuppose x, y ∈ C (with x = y ) have support. Then x + N C ( x ) ∩ y + N C ( y ) = φ . Proof.
In fact if the two translated normal cones had a non-void intersection,then for points in the intersection the minimum distance from C problem wouldhave a non unique solution. But this in view of the preceding Lemma is acontradiction and therefore our statement is proved. Remark 34
The last Theorem and Corollary show that the support conditioncan arguably be viewed as local. There may be a closed set that at a single point,or in a whole region, behaves locally like a convex set, although it is not convexat all.
The two above results imply the following:
Theorem 35
Consider a closed set C without interior and suppose the set sp ( C ) of all support points of C is such that the set AN F ( sp ( C )) covers C .Then the projection Theorem hold good for C . This line of arguing makes it possible to state the following generalizationof the Convexity Theorem for closed sets without interior.
Theorem 36
Consider a closed set C without interior. Then C is convex if andonly if the set sp ( C ) of all support points of C is such that the set AN F ( sp ( C )) covers C . And if this is the case C is equal to the intersection of the family ofclosed supporting semispaces of C . roof. The only if is an immediate consequence of the Projection Theorem.Conversely let Ψ be the intersection of semispaces defined in the statement ofthe Theorem, which is obviously a closed convex set containing C . Supposethere exists y ∈ Ψ such that y / ∈ C . Then by the preceding Theorem we canproject y on C . The unique projection z ∈ C has support because y − z is inthe normal cone to C at z and so ( y − z, w = z ) ≤ ∀ w ∈ C . But then thefunctional ( y − z, . ) leaves y in an open semispace opposite to the closed onecontaining C , thereby contradicting that y ∈ Ψ. Thus Ψ = C and we are done. For the case where C is bounded body in a Hilbert space, we will study convexnon-smooth hypersurfaces. We will prove, in our infinite dimensional and non-smooth setting, that there exists a homeomorphism of the boundary of anyconvex bounded body onto the boundary of the sphere . Thus the boundary ofany convex bounded body is what we will define to be a convex hypersurface.Our final Theorem in the Section will state a sort of converse of this result,showing that if we consider a convex bounded hypersurface, this is exactly theboundary of both its convex and its closed convex extension (assumed to haveinterior), and that these two sets are the same.We note that we will make no use of the weak topology and compactness,since weak topology seems to be of no help in the present matter. Also noticethat in our approach we have a global description of a hypersurface as the rangeof a non-linear function (the homeomorphism) defined on a set that we can fixto be the boundary of the unit sphere. In this sense we talk of a (non-linear)range space theory. We will comment at the end on the fact that a linear rangespace theory can also be used for the same purposes.We define now the closed hypersurface, where closed stands for ”withoutboundary”. The word closed will be tacitly understood in the sequel to avoidconfusion with topological closedness (and actually an hypersurface is assumedto be closed). On the other hand we will not deal here with hypersurfaces withboundary.In finite dimension the definition can assume that there is a one to one con-tinuous mapping on the surface of a sphere onto the hypersurface. Then thehypersurface is actually homeomorphic to the surface of the sphere for free, sincethere is a Theorem in [8], stating that a continuous one to one function on a com-pact set to a Hausdorff topological space has a continuous inverse. Thus in thepresent infinite dimensional case it seems natural to assume a homeomorphismfrom scratch.Also we will deal with the case where convex sets have interior and thisseems intuitively founded just in view of such homeomorphism to the boundaryof a convex body like the sphere. 15 efinition 37 A (Jordan) hypersurface Σ in a Hilbert space is a set which ishomeomorphic (in the strong topology sense) to the boundary of a closed sphere(and therefore it is a closed set). A hypersurface is convex if it has support ateach of its points. Our first aim is to show that if we consider a bounded convex body C thenits boundary B ( C ) is a bounded convex hypersurface. Theorem 38
Let C be a bounded convex body. Then its boundary of C is abounded convex hypersurface. Proof.
The proof is constructive in the sense that we will exhibit an appropriatehomeomorphism. First of all, as usual, we may assume that 0 ∈ C i and so wemay consider a closed sphere S r around the origin of radius say r >
0, entirelycontained in C i . Each ray ρ emanating from the origin meets obviously theboundary of S r (which we denote by Φ) in a unique point, but also meets B ( C )(which we denote by Ω) in another unique point thanks to Theorem 8. Nowdefine by ϕ the map that associates each point y of Φ with the unique point ϕ ( y ) which is the intersection of the ray generated by y with Ω. The map ϕ isobviously onto, since each point of Ω generates a ray on its own. It is also oneto one because distinct rays meet only at the origin, and so two different valuesof ϕ in Ω can only be generated by distinct vectors in Φ. Next we show thatthe map is continuous. In fact consider any point z ∈ Ω. We can define a baseof the neighborhood system of z , intersecting closed spheres around z with Ω.Consider one of those spheres Σ ε with radius ε >
0. Notice that C o (Σ ε ) ⊃ Σ ε .Now if we take the cone generated by a closed sphere around r z k z k of radius ε r k z k this cone will contain a sphere of radius ε/ z. This proves continuity of ϕ . On the other hand ϕ − ( z ) = z k z k , and since k z k is bounded from above andfrom below on Ω, ϕ − is continuous too. This completes the proof.Next we tackle the problem of stating a result going in the reverse direction,that is, from the convex hypersurface to the appropriate convex body. Theorem 39
Suppose Ω is a bounded convex hypersurface and assume that the(necessarily bounded) set C − (Ω) has interior. Then Ω = B ( C − (Ω)) Moreover, (in view of Corollary 9) C − (Ω) = C (Ω) and so also Ω = B ( C (Ω)) Proof.
We know from the preceding Theorem that B ( C − (Ω)) is a boundedconvex hypersurface and elementary computations in [1] show that C − (Ω) i is aconvex set. Furthermore, because each point of Ω is a support point for Ω itself16nd hence also for C − (Ω), it follows that Ω ⊂ B ( C − (Ω)). As usual we assumewithout restriction of generality that 0 ∈ C − (Ω) i . In order to simplify notationsin what follows, and again without restriction of generality, we assume that C − (Ω) i contains a closed sphere of radius 2 around the origin (if not we multiplyeverything by a positive scalar factor and nothing changes for the purposes ofthe present proof). From our previous theory we know that each ray emanatingfrom the origin meets B ( C − (Ω)) (and hence possibly Ω ⊂ B ( C − (Ω)) in a uniquepoint. Let ϕ be the homeomorphism of B ( S ) onto Ω. What we want to donow is to extend this homeomorphism to an homeomorphism on the whole of S onto the set [0 : 1]Ω. This will be done in two pieces: first we extend it to S \ S i and then we will take the identity map on S i . We call ψ the extendedmap. Bear in mind that M ≥ k z k ≥ m > ∀ z ∈ Ωfor some
M > m . Thus also: M ≥ k z k ≥ m > ∀ z ∈ C − (Ω)For y ∈ S \ S i to simplify notations we put: γ ( y ) = ϕ (2 y k y k )which is clearly a continuous function. Then on S \ S i , ψ is defined by: ψ ( y ) = ( k y k − γ ( y ) + (2 − k y k ) y k y k In this way when y ∈ B ( S ) or k y k = 2, we have that ψ ( y ) = γ ( y ). When y ∈ B ( S ) or k y k = 1, then ψ ( y ) = y . In between we have a convex combinationof these two vectors. In the rest of S , ψ is defined to be the identity map. Nowit is readily seen that ψ is one to one and the inverse function on ([0 : 1]Ω) \ S is: ψ − ( z ) = z k z k (1 + k z k − k γ ( z ) k − ψ and ψ − are continuous. Thus ψ is an homeomorphism and it maps S onto the set [0 : 1]Ω. Because its rangeis bounded, the image of S i , which is an open set containing the origin andwhose boundary is Ω, must be bounded. Thus each ray emanating from theorigin must meet Ω and so, in view of Corollary 9 it must be Ω = B ( C − (Ω)).This completes the proof . Remark 40
There is another sort of range space theory, developed in [2]) thatstill studies convex structures, but using the range of a linear (instead of non-linear) continuous functions, that is, of operators. We know that a convex closedset is the intersection of closed half spaces. Now in a separable Hilbert spaceenvironment (which may be represented by the space l ) it is well known that e can limit the intersection to a countable number of semispaces. Under mildassumption the corresponding countable set of inequalities takes on the form Lx ≤ v where L is an operator and v is a vector (called the bound vector). Therange space point of view consists at looking at this equation from the side ofthe of the range R ( L ) of L , instead of the side of the unknown x . Just to givea taste of it, from the range space point of view, this inequality has solutionif and only if v ∈ R ( L ) + P , where P is the closed pointed cone of all non-negative (componentwise) vectors (which, incidentally has void interior!). Notethat R ( L )+ P is a cone albeit a non pointed one. With this range space approachan extensive theory of geometry, feasibility, optimization and approximation hasbeen developed. For details, proofs and more the interested reader may look atthe paper cited above. Intuitively, what is meant here by convexification of a bounded hypersurface,can be illustrated colloquially and in three dimension by this image: wrap anon-convex bounded hypersurface with a plastic kitchen pellicle and the thenpellicle takes the form of a surface, which is the convexification of the originalsurface. Here is a formal definition
Definition 41
Consider a closed bounded hypersurface Φ in a a real Hilbertspace and assume that C − (Φ) = C (Φ) (where the equality follows from Corollary9) has non-void interior. Then Ω = B ( C (Φ)) , which by Theorem 38 is a boundedconvex hypersurface, is called the convexification of Φ . Naturally the convexification of a bounded convex hypersurface satisfyingthe above condition is the surface itself. So to speak, convexity is the fixedpoint of the convexification operator.A straightforward consequence of the work carried out so far is the following:
Theorem 42
Let Φ be a closed bounded hypersurface in a real Hilbert spaceand let Ω its convexification. Then Φ is homeomorphic to Ω . Proof.
Lets indicate by the symbol ∼ the relation ”is homeomorphic to” andby S the closed unit sphere around the origin. Then we know that, by definition,Φ ∼ B ( S ). But by Theorem 38 we also know that Ω ∼ B ( S ). Because ∼ is anequivalence relation, it follows also Ω ∼ Φ.The above (and recurring) hypothesis that C (Φ) has interior is not neededin finite dimension. In fact we state the following Theorem. By the way, recallthat in finite dimension, from the only fact that Φ is compact it follows that C (Φ) is compact too. Theorem 43
Let Ω be a compact convex hypersurface in R n . Then A (Ω) = R n ,where A (Ω) is the affine extension of Ω . Thus C (Ω) is a convex body. roof. In finite dimension every convex set has relative interior. Thus we canargue momentarily in E = L ( C (Ω)), where C (Ω) has interior. In this space, inview of Theorem 39 we can say that:Ω = B ( C − (Ω)) C − (Ω) = C (Ω)and Ω = B ( C (Ω))Next, as we did before we extend the homeomorphism ϕ of B ( S ) onto Ω to thewhole of S . Thus Π = ψ ( S i ) is an open set and Ω is its boundary. We intersecteach coordinate axis with Π to obtain a relatively open subset and hence asegment of the form [0 : z i ) with z i = 0 and laying on the ith coordinate axis.But then z i is in the boundary of Π, that is, in Ω. Thus A (Ω) = R n and so weestablished our thesis. References [1] J.L. Kelley, I. Namioka et al, ”Linear Topological Spaces” , Graduated textin Mathematics, Springer, New York 1963[2] P. d’Alessandro, ”Optimization and Approximation for Polyhedra in Sepa-rable Hilbert Spaces” , AJMAA Vol 12, Issue 1,Article 7,pp 1-27,May 2015.[3] R. R. Phelps,
Lectures on Choquet Theorem , Springer, New York, 2001[4] J. Stoer, C. Witzgall,
Convexity and Optimization in finite dimensions I ,Springer-Verlag Berlin-Heidelberg-New York 1970.[5] P. d’Alessandro -
Closure of Pointed Cones and Maximum Principle inHilbert Spaces , CUBO A Mathematical Journal, Vol 13 No 2, June 2011[6] Leo Jonker,
Hypersurfaces of Nonnegative Curvature in A Hilbert Space,
Transactions of the American Mathematical Society, Vol. 169 (Jul., 1972),pp. 461-474[7] P. d’Alessandro -