Generalization of Klain's Theorem to Minkowski Symmetrization of compact sets and related topics
aa r X i v : . [ m a t h . M G ] A ug Generalization of Klain’s Theorem to MinkowskiSymmetrization of compact sets and related topics
Jacopo Ulivelli ∗ Abstract
We shall prove a convergence result relative to sequences of Minkowski symmetrals ofcompact sets. In particular, we investigate the case when this process is induced by sequencesof subspaces whose elements belong to a finite family, following the path marked by Klain in[11], and the generalizations in [4] and [2].We prove an analogue result for Fiber symmetrization of a specific class of compact sets,namely the convex shells. The idempotency degree for symmetrization of this family of setsis investigated, leading to a simple generalization of a result from Klartag [12] regarding theapproximation of a ball through a finite number of symmetrizations.Two counterexamples to convergence of sequences of symmetrals in the plane are proven,extending some ideas in [2] to a wider class of i -symmetrizations, which include the Minkowskione. Steiner symmetrization has been introduced in attempting to prove the isoperimetric inequality forconvex bodies in R n . Its most useful feature is that there are sequences of hyperplanes such thatthe corresponding successive symmetrals of a convex body always converges to a ball. Nowadaysthis property is employed in standard proofs of not only the isoperimetric inequality but also ofother potent geometric inequalities, like the Brunn-Minkowski, Blascke -Santalò or Petty projectioninequality. Recently its role has been crucial in the solution of a long due open problem aboutAffine Quermassintegrals [15]. Other symmetrizations, like Minkowski and Schwarz satisfy a similarproperty.Let us introduce some terminology. Let E be the class K nn of convex bodies in R n or the class C n of compact sets in R n . Given a subspace H ⊂ R n let ♦ H denote a symmetrization over E , i.e. amap which associates to every set in E a set in E symmetric with respect to H . Given a sequence { H m } of subspaces and K ∈ E we define the sequence K m = ♦ H m . . . ♦ H ♦ H K. For which sequences { H m } and for which symmetrizations ♦ H the sequence { K m } converges foreach K ∈ E ? This process depends on the class E , on the definition of ♦ H and on the sequence { H m } (and, in particular, on the dimension of the subspaces). This research belongs to a serieswhich is trying to better understand the convergence of this process.The cases which have been studied most are those of Steiner, Schwarz and Minkowski symmetriza-tions in the class K nn and for symmetrizations with respect to hyperplanes but some results areavailable also for more general symmetrizations, for the class of compact sets and for subspacesof any dimension. See, for instance Klartag [12], Coupier and Davydov[7], Volcic [19], Bianchi,Gardner and Gronchi [4] and the very recent [1] from Asad and Burchard.We start from the analysis and extension of a counterexample from [2], where is proven thatfor suitable sequences of directions it is not possible to achieve convergence for the correspondingsequence of Steiner symmetrals of compact sets, provided that the set is chosen to have certainproperties. In the first generalization we prove that the impossibility of convergence depends onlyon the presence of a perpetually spinning segment contained in the sequence of symmetrals. Inthe second one we use a technical result from [3] to extend the same idea to a wider class ofsymmetrizations. ∗ Email: [email protected]fi.it
Theorem 1.1 (Klain) . Given K ∈ K nn and a finite family F = { Q , ..., Q l } ⊂ G ( n, n − , considera sequence of subspaces { H m } m ∈ N such that for every m ∈ N , H m = Q j for some ≤ j ≤ l . Thenthe sequence K m := S H m ...S H K converges to a body L ∈ K nn . Moreover, L is symmetric with respect to Q j for every Q j whichappears infinitely often in the sequence. This result has been vastly extended in [4]. In particular it holds for Minkowski symmetrization,Fiber symmetrization and Minkowski-Blaschke symmetrization. We will properly define these andother concepts in the next section. In [2] it is proven a generalization of Theorem 1.1 for the Steinersymmetrization of compact sets, and our goal will be to prove the same for Minkowski and Fibersymmetrizations, partially answering a question posed in [4].In Section 4, after observing that we lose the properties of being idempotent when passing fromconvex to compact sets, we prove a first result regarding the iteration of the same Minkowskisymmetrization over a compact set. We use the ideas in this result and the Shapley-Folkman-Starr Theorem (see for example [18] for Starr’s version, or [17] for a complete development of thesubject) to prove Klain’s result for the Minkowski symmetrization of compact sets, which is a directconsequence of our main result.
Theorem 1.2.
Let K be a convex compact set and let { H m } be a sequence of subspaces of R n (notnecessarily of the same dimension) such that the sequence of iterated symmetrals K m := M H m ...M H K converges to a convex compact set L in Hausdorff distance. Then the same happens for everycompact set ˜ K such that conv ( ˜ K ) = K , and the sequence ˜ K m , defined as ˜ K m := M H M . . . M H ˜ K ,converges to the same limit L . In Section 5 we introduce the concept of convex shell , a generalization of the more known convexannulus . We say that a set is a convex shell if it is the difference between a convex compact set L and an open set whose closure is contained in the interior of L . We exploit the properties of thisobjects of having positive measure and convex outer boundary to prove some results regarding theexistence of a degree of idempotency for Minkowski and Fiber symmetrizations depending only onthe body, and not on the dimension of the space. By these means we provide some characterizationsof invariance under symmetrization for this kind of sets. We conclude proving Klain’s Theoremfor compact sets with convex outer boundary and positive measure. Theorem 1.3.
Let K be a compact set such that ∂ conv ( K ) ⊂ K and | K | > , F = { Q , ..., Q s } afamily of subspaces of R n , and { H m } a sequence such that { H m } ∈ F for every m ∈ N . Then thesequence K m := F H m ...F H K converges to a convex set L . Moreover L it is the limit of the same symmetrization process appliedto conv ( K ) , and it is symmetric with respect to all the subspaces of F which appear infinitely oftenin { H m } . As usual, S n − denotes the unit sphere in the Euclidean n -space R n with Euclidean norm k·k .The term ball will always mean an n -dimensional euclidean ball, and the unit ball in R n will bedenoted B n . B ( x, r ) is the ball with center x and radius r . If x, y ∈ R n , we write x · y for the innerproduct. If x ∈ R n \ { o } , then x ⊥ is the ( n − x . G ( n, i )denotes the Grassmanian of the i -dimensional subspaces of R n , 1 ≤ i ≤ n −
1, and if H ∈ G ( n, i ), H ⊥ is the ( n − i )-dimensional subspace orthogonal to H . By subspace we mean linear subspace .Given x ∈ R , ⌊ x ⌋ is the floor function of x .If X is a set, we denote by conv X its convex envelope, and ∂X its boundary. If H ∈ G ( n, i ),then X | H is the (orthogonal) projection of X on H . If X and Y are sets in R n and t ≥
0, then tX := { tx : x ∈ X } and X + Y := { x + y : x ∈ X, y ∈ Y } Minkowski sum of X and Y . For X measurable set, its volume in the respectivedimension will be | X | .When H ∈ G ( n, i ) , we write R H for the reflection of X in H , i.e. the image of X under the mapthat takes x ∈ R n to 2( x | H ) − x , where x | H is the projection of x onto H . If R H X = X , we saythat X is H -symmetric.We denote by C n the class of nonempty compact subsets of R n . K n will be the class of non emptycompact convex subsets of R n and K nn is the class of convex bodies , i.e. members of K n with interiorof positive measure. In the same way we define C nn . If K ∈ K n , then h K ( x ) := sup { x · y : y ∈ K } , for x ∈ R n , defines the support function h K of K . With the support function we can define themean width of a convex body K , which is w ( K ) := 1 | ∂B n | Z S n − ( h K ( ν ) + h K ( ν )) dν := R − S n − ( h K ( ν ) + h K ( ν )) dν. If X is a measurable set such that | X | > f : X → R is a measurable function, notationwisewe have R − X f ( ξ ) dξ := 1 | X | Z X f ( ξ ) dξ. The aforementioned spaces C n and K n are metric spaces with the Hausdorff metric , which is givenin general for two sets
A, B by d H ( A, B ) := sup { e ( A, B ) , e ( B, A ) } , where e ( A, B ) := sup x ∈ A d ( x, A )is the excess of the set A from the set B , and d ( x, A ) is the usual distance between a point and aset. The completeness of such metric spaces is a classic result [6], we will refer to it as Blaschkeselection Theorem both in convex and compact context.Another classical result we will refer to is the Brunn-Minkowski inequality. Given two compactsets
A, B , it states that | A + B | /n ≥ | A | /n + | B | /n , where equality holds if and only if A is convex and B is a homothetic copy of A (up to subsets ofvolume zero).Given C ∈ C n , H ∈ G ( n, i ) , ≤ i ≤ n −
1, we recall the definition of some symmetrizations: • Schwarz symmetrization: S H C := [ x ∈ H B ( x, r x ) , where r x is such that | B ( x, r x ) | = (cid:12)(cid:12) C ∩ ( H ⊥ + x ) (cid:12)(cid:12) , and B ( x, r x ) ⊂ H ⊥ + x . If (cid:12)(cid:12) C ∩ ( H ⊥ + x ) (cid:12)(cid:12) =0 then r x = 0 when C ∩ ( H ⊥ + x ) = ∅ , while when the section is empty then the symmetriza-tion keeps it empty.For i = n − ≤ i ≤ n − Schwarz symmetrization . • Minkowski symmetrization: M H C := 12 ( C + R H C ) . We will also consider the case i = 0, which is called the central Minkowski symmetrization M o K = K − K . • Fiber symmetrization: F H C := [ x ∈ H (cid:20) (cid:0) C ∩ ( H ⊥ + x ) (cid:1) + 12 (cid:0) R H C ∩ ( H ⊥ + x ) (cid:1)(cid:21) . M H ⊥ ,x the central Minkowski symmetrization with respect to x in H ⊥ + x identified with R n − i , we can write F H K = [ x ∈ H M H ⊥ ,x ( K ∩ ( H ⊥ + x )) . • Minkowski-Blaschke symmetrization: If K is a convex compact set we define h M H K ( u ) := R − S n − ∩ ( H ⊥ + u ) h K ( v ) dv, if (cid:12)(cid:12) S n − ∩ ( H ⊥ + u ) (cid:12)(cid:12) = 0 in R n − i h K ( u ) , otherwise.At the end of Section 4 we will see that we can extend this definition to any compact setusing the support function of its convex envelope.Consider a family of bodies B and a subspace H ∈ G ( n, i ), then an i -symmetrization is a map ♦ H : B → B H , where B H are the H -symmetric elements of B .We state for later use some properties of i -symmetrizations. Consider K, L ∈ B , H a subspace in R n , then we have: Monotonicity : K ⊂ L ⇒ ♦ H K ⊂ ♦ H L ; H-symmetric invariance : R H K = K ⇒ ♦ H K = K ; H-orthogonal translation invariance for H-symmetric sets : R H K = K, y ∈ H ⊥ ⇒ ♦ H ( K + y ) = ♦ H K .When this three properties hold, we have the following result from [3]. Lemma 2.1.
Let H ∈ G ( n, i ) , ≤ i ≤ n − , B = K n or K nn . If ♦ is a i -symmetrization suchthat it has the properties of monotonicity, H-symmetric invariance and H-orthogonal translationinvariance for H-symmetric sets, then F H K ⊂ ♦ H K ⊂ M H K for every K ∈ B . Notice that these properties hold for Steiner, Minkowski and Fiber symmetrizations, while thefirst and the third hold for Schwarz symmetrization.
Example 3.1.
We first present and comment an example from [2].
In [2] it is proved that for certain kind of sequences of directions in the plane it is possible toconstruct a compact set K such that the sequence of iterated Steiner symmetrals induced by thosedirections does not converge. These sequences are built as follows.Consider a sequence of angles { α m } ⊂ (0 , π/
2) such that X m ∈ N α m = + ∞ , X m ∈ N α m < + ∞ , (1)and take the further sequence of directions given by u m := (cos β m , sin β m ) , where β m := m X j =1 α j . < γ := Q m ∈ N cos α m and let U i := span( u i ), if we consider a compact set K with area0 < | K | < π ( γ/ and containing a horizontal unitary segment ℓ centered in the origin, thesequence of compact sets K m := S U m ...S U K doesn’t converge.The main idea behind this example is that the sequence of directions { u m } , which corresponds tothe directions of the projections ℓ m := K m − | U m = K m ∩ U m is dense in S . In fact K ⊃ ℓ , K ⊃ ℓ , and so on for the monotonicity of Steiner symmetrization.Thus the the sequence { ℓ m } is perpetually counterclockwise spinning around the origin, and thelength of ℓ m always exceeds γ . Now, if a limit exists for K m , it must contain a ball of diameter γ ,but this is a contradiction because | K | < π ( γ/ .In the next example we see that what really matter are the rotations of a suitable sequence ofsegments, like { ℓ m } . Example 3.2.
Now we observe what happens when our symmetrizations are close to π/ , whilein the previous one the angles were close to .Proof. Consider a sequence { α m } ⊂ (0 , π/
2) of angles with the properties (1) in Example 3.1.With it we build the sequence ν m := π m X j =1 α m , and a corresponding sequence of directions u m := (cos ν m , sin ν m ). This corresponds to a processwhere the rotating frame of reference is cos m − X j =1 α m , sin m − X j =1 α m , cos π/ m − X j =1 α m , sin π/ m − X j =1 α m , which are respectively the horizontal and vertical axes, and at the m -th iteration the new sym-metrization will exceed the rotated vertical axis of α m degrees.Consider an ellipse E with a horizontal unitary segment ℓ as larger diameter, centered in theorigin and with axes lying on the directions of the orthogonal frame of reference. Moreover werequire that | E | < π ( δ/ , where δ := Y m ∈ N cos α m > . We start observing what happens for a single symmetrization in a direction u := (cos α, sin α ) , α ∈ ( π/ , π ). Applying the symmetrization S U , U := span { u } , to E , for the monotonicity of Steinersymmetrization we have S U ℓ ⊂ S U E. Moreover, as we prove in the following Lemma, (cid:12)(cid:12) U ⊥ ∩ S U E (cid:12)(cid:12) = (cid:12)(cid:12) U ⊥ ∩ E (cid:12)(cid:12) ≥ sin α. We recall to the reader that the Steiner symmetrization of an ellipse is still an ellipse, and thatthe axes of S U E lay on U and U ⊥ . Lemma 3.3.
Choose an orthonormal basis { e , e } , take u = (cos α, sin α ) , U = span ( u ) and anellipse E such that its axes with semilengths a, b lay on e and e respectively. Then, if a ′ and b ′ are the semilengths of the axes of the ellipse S U E , with a ′ := (cid:12)(cid:12) U ⊥ ∩ S U E (cid:12)(cid:12) , then a ′ ≥ a sin α. Proof.
We know that S U E is still an ellipse with axes laying in U and U ⊥ , and by definition a ′ ishalf of the length of the section (cid:12)(cid:12) U ⊥ ∩ S U E (cid:12)(cid:12) . In particular, we can see a ′ as the norm of the vector( a cos( π/ − α ) , b sin( π/ − α )), or, equivalently, ( a sin α, b cos α ). Thus( a ′ ) = a sin α + b cos α ≥ a sin α, concluding the proof. 5or our purpose we can consider α = π/ α , ¯ α ∈ (0 , π/ α = cos ¯ α .Calling U j := span ( u j ), for the m -th symmetral E m := S U m ...S U K we obtain the inequality | ℓ m | := (cid:12)(cid:12) U ⊥ m ∩ E m (cid:12)(cid:12) ≥ m Y j =1 cos α j , ℓ m ⊂ E m , α j ∈ (0 , π/ , j = 1 , ..., m. In general S U m rotates ℓ m − counterclockwise of α m degrees, contracting it by a factor cos α m .We have that P m ∈ N ( α m ) < + ∞ , thus δ >
0, implying that every symmetral contains a segmentof length δ centered in the origin as a subset of ℓ m . This segment spins indefinitely counter-clockwise, because P m ∈ N α m diverges, and | ℓ m | always exceeds δ for the monotonicity of Steinersymmetrization. If we consider the sequence of the directions { ℓ m } , we observe that it is dense in S , thus it can approximate every direction with the limit of one of its subsequences. Thus, if alimit exists for E m , it must contain all these diameters, and with that a ball of diameter δ . Butwe chose E such that | K | < π ( δ/ , which gives us a contradiction.This example can be easily extended to every compact set C such that E ⊂ C , where E isagain an ellipse with a unitary diameter and | C | < π ( δ/ , thanks to the monotonicity of Steinersymmetrization.We can create new sequences of this kind combining these two examples. Notationwise, { α m } is the sequence in Example 3.1, { ˜ α m } the one in Example 3.2 used to build the sequence ν m = π/ P mj =1 ˜ α m . We can combine these two sequences as follows: ξ m = ξ m − + α, where α can be in { α m } or { ˜ α m } . The corresponding directions of symmetrization will be u m := ( (cos ξ m , sin ξ m ) if α was in { α m } ,(cos ξ m , sin ξ m ) ⊥ otherwise . Here ξ = 0 and the "rotation zero" is supposed to be in { α m } . Then we set ǫ := ∞ Y m =1 cos( α m ) cos( ˜ α m ) , replacing the previous values of γ and δ . In the hypothesis of Example 3.2 we now have our mixedcounterexample following the same steps, except the fact that now our conditions for the sequence { ξ m } become | ξ m | = + ∞ and X m ∈ N α m < + ∞ , X m ∈ N ˜ α m < + ∞ . Remark.
The subtle similarity between Example 3.1 and Example 3.2 lays on the behavior ofthe sequence of segments { ℓ m } . In fact, in both cases the sequence of the directions of the segmentsis of the type described in (1). In the former case this is immediate, because ℓ m lays always onthe axis of symmetrization, while in the latter { ℓ m } lays on the direction orthogonal to the axis ofsymmetrization. Being ν m = π/ P ni =1 α m , { α m } is again the sequence of the directions of thesegments { ℓ m } . Example 3.4.
We will now prove that similar counterexamples hold for symmetrizations whichsatisfy the hypothesis of Lemma 2.1. In particular, this holds for the Minkowski symmetrization.Proof.
Consider a set K ∈ K such that it contains a unitary horizontal segment and with meanwidth 1 / (2 π ) < w ( K ) < γ , where gamma is as in the example from [2]. In the hypothesis ofLemma 2.1 we have that S U m ...S U K ⊆ ♦ U m ... ♦ U K ⊆ M U m ...M U K, again U j := span( u j ), and we used Steiner symmetrization because it is equivalent to Fiber sym-metrization relative to a hyperplane, which is our case working on R .6 ⊥ V K V ⊥ V F V K Figure 1In this way we can exploit the first counterexample and the inclusion chain of Lemma 2.1 toguarantee that, if a limit exists for ♦ U m ... ♦ U K and M U m ...M U K , reasoning as before it mustcontain a ball of diameter γ , therefore this limit must have mean width greater than γ . In particularthis holds for the sequence of Minkowski symmetrals. But Minkowski symmetrization preservesmean width, that we supposed to be less than γ . This is a contradiction, thus there cannot be alimit. Two of the main features of Steiner, Schwarz, Minkowski and Fiber symmetrizations are the idem-potency and the invariance for H -symmetric bodies in the class of convex sets. These two propertiesdo not longer hold when we switch to the class of generic compact sets.An immediate example regarding Minkowski symmetrization is the following. Consider in R thecompact set C = { ( − , , (1 , } . This set is obviously symmetrical with respect to the verticalaxis, which we can identify with a subspace H . Then we have M H C = { ( − , , (0 , , (1 , } , thus the invariance for symmetric sets does not longer hold. If we apply again the same sym-metrization, M H ( M H C ) = { ( − , , ( − / , , (0 , , (1 / , , (1 , } , showing that the same happens to idempotency. In Figures 1 and 2 we see an example for theFiber symmetrization of a compact set in the plane.If we iterate this process for C = { ( − , , (1 , } , we see that in this case there is not a finitedegree of idempotency, i.e. do not exist an index ℓ ∈ N such that M ℓH C = M k + ℓH C for every k ∈ N , where in general M H . . . M H | {z } ℓ -times := M ℓH . Moreover the iterated symmetrals converge to the set given by conv( C ). This is the main ideabehind the next result, after proving a technical Lemma. Lemma 4.1.
Let K ∈ C n , H a subspace of R n . Theni) for every v ∈ R n M H ( K + v ) = M H ( K ) + v | H, ii) if K is H -symmetric, then K ⊆ M H K ,iii) K = M H K if and only if K is convex and H -symmetric. ⊥ V F V K V ⊥ V F V K Figure 2
Proof.
The first statement follows from the explicit calculations M H ( K + v ) = K + v + R H ( K + v )2 = K + R H ( K )2 + v | H ⊥ + v | H − v | H ⊥ + v | H M H ( K )+ v | H, where we used the linearity of the reflections and the decomposition v = v | H + v | H ⊥ .For the second statement, by hypothesis we have that R H K = K , i.e. R H ( x ) ∈ K for every x ∈ K . Then, taking x ∈ K , ( x + R H ( R H ( x ))) / x ∈ M H K , concluding the proof.Consider now K such that K = M H K . Then obviously K must be H -symmetric, and K = R H K .Then, for every x, y ∈ K we have that ( x + y ) / ∈ K , thus for every point z in the segment [ x, y ]we can build a sequence by bisection such that it converges to z . K is compact, henceforth itcontains z . The other implication is trivial.Notice that the second statement implies that K m ⊆ K m +1 for every m ∈ N . Theorem 4.2.
Let K ∈ C n , H ∈ G ( n, i ) , ≤ i ≤ n − . Then the sequence K m := M mH K = M H ...M H | {z } m-times K converges in Hausdorff distance to the H -symmetric convex compact set L = conv ( M H K ) . Proof.
We observe preliminarly that for the properties of convex envelope and Minkowski sum wehave K m ⊆ L for every m ∈ N . Then we only need to prove that for every x ∈ L we can find asequence x m ∈ K m such that x m → x . We can represent K as ¯ K + v, v ∈ K , where ¯ K contains theorigin. Being Minkoswki symmetrization invariant under H -orthogonal translations, we can take v ∈ H .For every m we have R H K m = K m , and thus we can write K m +1 = M H K m = K m + K m K + ... + K m . Considering the aforementioned representation of K , R H K = R H ¯ K + v , and we have K m = ¯ K m + v, where ¯ K m := M mH ¯ K, thus we can write every point y ∈ K m as y = ¯ y + v, ¯ y ∈ ¯ K m .Given x ∈ L , thanks to Carathéodory Theorem there exist x k ∈ K , λ k ∈ (0 , , k = 1 , ..., n + 1such that P n +1 k =1 λ i = 1 and x = n +1 X k =1 λ k x k = n +1 X k =1 λ k ¯ x k + v, x k = ¯ x k + v, ¯ x k ∈ ¯ K . For every λ k we consider its binary representation λ k = + ∞ X ℓ =1 a ℓ,k ℓ , a ℓ,k ∈ { , } (we do not consider ℓ = 0 because λ i < m -th approximation given by the partial sum λ m,k := m X ℓ =1 a ℓ,k ℓ = 12 m m X ℓ =1 a ℓ,k m − ℓ . We notice for later use that | λ k − λ m,k | ≤ / m .Calling q s := ⌊ s / ( n + 1) ⌋ we now build the sequence x s := n +1 X k =1 λ q s ,k ¯ x k + v = 12 q s n +1 X k =1 q s X ℓ =1 a ℓ,k q s − ℓ ! ¯ x k + v, where the 2 s + ν − q s ( n + 1) spare terms in ¯ K can be taken as the origin in the sum representing¯ K s .Then we have that every x s belongs to K s , and k x − x s k = k ¯ x + v − (¯ x s + v ) k ≤ n +1 X k =1 k ¯ x k k| λ k − λ q s ,k | ≤ q s n +1 X k =1 k ¯ x k k ≤ ( n + 1) max y ∈ K k y − v k q s . Clearly k x − x s k → s → + ∞ , which concludes our proof.As immediate consequence we have the following result. Corollary 4.3.
In the hypothesis of Theorem 4.2, we have that the sequence K m := F mH K = F H ...F H | {z } m-times K converges in Hausdorff distance to the H -symmetric compact set L = [ x ∈ H conv ( F H K ∩ ( x + H ⊥ )) . Proof.
Recalling the definition of Fiber symmetrization F H K = [ x ∈ H
12 (( K ∩ ( x + H ⊥ )) + ( R H K ∩ ( x + H ⊥ ))) = [ x ∈ H M H ( K ∩ ( x + H ⊥ )) . The result is a straightforward application of Theorem 4.2 to the sections of K . Remark
In Corollary 4.3 we lose the convexity on the limit, but there still holds convexity forits sections, as a consequence of Theorem 4.2. This property is known, when dim( H ) = 1, as directional convexity (see [14]). We can extend this concept to sectional convexity , that is, fixed asubspace H in R n and a set A , the convexity of every section A ∩ ( x + H ) , x ∈ H ⊥ . Then in theprevious result the sectional convexity is with respect to the subspace H ⊥ .We now state Shapley-Folkman-Starr Theorem ([18],[17]) for using it in the next proof. Theorem 4.4 (Shapley-Folkman-Starr) . Let A , ..., A k ∈ C n . Then d H ( k X j =1 A j , conv ( k X j =1 A j )) ≤ max ≤ j ≤ k D ( A j ) , where D ( · ) is the diameter function D ( K ) := sup {k x − y k : x, y ∈ K } . Following the idea of Theorem 4.2 and the formula given by Shaple-Folkman-Starr Theorem, weobtain our main result. 9 roof of Theorem 1.2.
We will show that the theorem holds proving that d H ( ˜ K m , K m ) → m → ∞ .We can write K m as the mean of Minkowski sum of composition of reflections of K . In fact wehave K = K + R H K ,K = K + R H K + R H ( K + R H K )4 = K + R H K + R H K + R H R H K ,... and so on. The same obviously holds for ˜ K m . Calling these reflections A j , ≤ j ≤ m , and A j := A j ˜ K we can write ˜ K m = 12 m m X j =1 A j ˜ K = 12 m m X j =1 A j . Now, the convex envelope commute with Minkowski sum and isometries, thusconv ˜ K m = 12 m m X j =1 A j conv( ˜ K ) = 12 m m X j =1 A j K = K m , and using the Shapley-Folkman-Starr Theorem we obtain the estimate d H ( ˜ K m , K m ) = d H m m X j =1 A j , m conv( m X j =1 A j ) ≤ √ n m max ≤ j ≤ m D ( A j ) . The sets A j are all isometries of K , thus D ( A j ) = D ( ˜ K ), which is finite, completing the proof.We observe that the example given at the beginning of this section gives us a proof of the factthat the upper bound for the convergence rate is sharp. In fact, it’s easy to check that for thecompact set C = { ( − , , (1 , } and the segment L = conv( C ) we have d H ( C m , L ) = 12 m | L | . We now have, as a consequence of Theorem 1.2, our generalization for Klain’s result.
Corollary 4.5.
Let K ∈ C n , F = { Q , ..., Q s } ⊂ G ( n, i ) , ≤ i ≤ n − , { H m } a sequence ofelements of F . Then the sequence K m := M H m ...M H K converges to a convex set L such that it is the limit of the same symmetrization process appliedto ¯ K = conv ( K ) . Moreover, L is symmetric with respect to all the subspaces of F which appearinfinitely often in { H m } .Proof. The proof follows straightforward from the generalization of Klain Theorem to Minkowskisymmetrization of convex sets in [3] and Theorem 1.2.We can use a similar method to generalize this classical result from Hadwiger, see for example[17].
Theorem 4.6. [Hadwiger] For each convex body K ∈ K nn there is a sequence of rotation means of K converging to a ball. In fact we can state Theorem 1.2 in a more general fashion:
Theorem 4.7.
Consider K ∈ K n and a sequence of isometries A m . If the sequence K m = 1 m m X j =1 A j K converges, then the same happens for every compact set C ∈ C n such that conv ( C ) = K , and thelimit is the same. Corollary 4.8.
For each compact set C such that conv ( C ) ∈ K nn there is a sequence of means ofisometries C converging to a ball. Remark.
Theorem 1.2 gives us an answer regarding the possibility of extending the Minkowski-Blaschke symmetrization M H to compact sets. This symmetrization that we have defined inSection 2 for convex bodies can be practically seen as the mean of rotations of a compact set K ∈ K n by a subgroup of SO ( n ), thus can be approximated by1 N N X k =1 A k K, where { A k } Nk =1 ⊂ { A k } k ∈ N a suitable set of rotations dense in said subgroup.In fact, from the definition of M H in terms of the support function, we have that the integral canbe approximated by N X k =1 h K ( A ∗ k x ) N = 1 N N X k =1 h A k K ( x ) , which corresponds naturally to the Minkowski sum written above.Then again, following the proof of Theorem 1.2, we can write the symmetral as the limit of a meanof Minkowski sum of isometries of a fixed K ∈ K n , and thus Minkowski-Blaschke symmetrizationactually gives the same result for every C ∈ C n such that conv( C ) = K .This shows that this symmetrization is sensible only to the extremal points of a set, thus there isno difference in using it with compact sets or convex sets. One of the main properties of Minkowski symmetrization is that, as a consequence of Brunn-Minkowski inequality, it strictly increases the volume of the symmetral. In fact, for every compactset K ⊂ R n such that | K | >
0, we have | M H K | /n = | / K + R H K ) | /n ≥ | K | /n + 12 | R H K | /n = | K | /n , where equality holds if and only if K is convex and R H K is homothetic to K (up to sets of measurezero), that is K is convex and H -symmetric. This happens if and only if K = M H K , thus wewould like to state that the iteration of Minkowski symmetrization increases the volume until thesequence of symmetrals reaches M H conv( K ).With Theorem 4.2 we proved that, regardless the volume, the limit of ˜ K m is actually M H conv( K ),but now we raise one more question: can we obtain this limit in a finite number of iterations? Underwhich hypothesis is this possible?We start by giving an answer for compact sets of R . Later, in Proposition 5.5, we prove that M H and the Fiber symmetrization have a finite degree of idempotency when the compact set belongsto a certain class. Lemma 5.1.
Let K ∈ R be a compact set such that conv ( K ) = [ a, b ] with the following property: ∃ ε > s.t. [ a, a + ε ] ⊂ K or [ b − ε, b ] ⊂ K. Then there exists an index ℓ ∈ N depending on ε and ( b − a ) / such that M ℓo K = M ℓ + ko K for every k ∈ N .Moreover, ℓ increases with ( b − a ) / and decreases if ε increases.Proof. First consider the case K ⊇ { a } ∪ [ b − ε, b ]. Then M o K ⊇ M o ( { a } ∪ [ b − ε, b ]) = (cid:20) a − b , a − b ε (cid:21) ∪ (cid:20) b − a − ε , b − a (cid:21) . K ⊇ [ a, a + ε ] ∪ { b } . Then, naming M := b − a , m := b − a − ε , and relabeling ε/ ε , we can work with a set containing a subset the form[ − M, − m ] ∪ [ m, M ] =: ˜ K, where M − m = ε .If now we apply the symmetrization, we obtain M o K ⊇ [ − M, − m ] ∪ (cid:20) m − M , M − m (cid:21) ∪ [ m, M ] =: ˜ K . If ( M − m ) / ≥ m , that is m ≤ M/
3, then M o K = conv( K ), and the result holds with ℓ = 1.In the general case we can show by induction that holds the inclusion M k +1 o K ⊇ ˜ K k +1 := M k +1 o ˜ K ⊇ k +1 [ j =0 (cid:20) (2 k +1 − j ) m − jM k +1 , (2 k +1 − j ) M − jm k +1 (cid:21) , where the first inclusion is trivial thanks to the monotonicity of Minkowski symmetrization. Inparticular we will show that˜ K k +1 ⊇ ˜ K k ∪ k [ j =1 (cid:20) (2 k +1 − j + 1) m − (2 j − M k +1 , (2 k +1 − j + 1) M − (2 j − m k +1 (cid:21) , which will contain the desired set. This inclusion is actually an equality, but proving this fact isbeyond our goal here. We leave it to the keen readers.For k = 1 we have already seen that the inclusion is true. By inductive hypothesis, at the k + 1-thstep the means of adjacent intervals of M ko ˜ K is given by12 (cid:26)(cid:20) (2 k − ( j + 1)) m − ( j + 1) M k , (2 k − ( j + 1)) M − ( j + 1) m k (cid:21) + (cid:20) (2 k − j ) m − jM k , (2 k − j ) M − jm k (cid:21)(cid:27) = (cid:20) (2 k +1 − j + 1) + 1) m − (2( j + 1) − M k +1 , (2 k +1 − j + 1) + 1) M − (2( j + 1) − m k +1 (cid:21) for every j = 0 , ..., k −
1, giving us the elements of the union with odd indices.Observe then that by inductive hypothesis ˜ K k is invariant under reflection. Thus, thanks toLemma 4.1 and the monotonicity of Minkowski symmetrization, we have ˜ K k ⊆ M k +1 o K , anddoubling both the terms over and under the fractions representing the extremal points of thesubintervals, we obtain the elements with even indices, concluding the induction.Taking at the k -th step two adjacent intervals, we have that they are connected if(2 k − ( j + 1)) M − ( j + 1) m k ≥ (2 k − j ) m − jM k . It follows that the condition for filling the whole segment conv( M kH K ) is mM ≤ k − k + 1 . Observe that the dependence on the index j has disappeared after the calculations, confirmingthat this holds for every couple of adjacent intervals.By hypothesis M − m = ε , and (2 k − / (2 k + 1) →
1. We have mM = 1 + m − MM = 1 − εM , then there exists ℓ ∈ N such that 1 − εM < ℓ − ℓ + 1 , thus M ℓo K = conv( K ) for ℓ ≥ log (cid:18) Mε − (cid:19) . This set is convex and o -symmetric, thus is invariant under Minkowski symmetrization. Thedependence from M and ε is clear from the last inequality.12 emark. This Lemma holds more in general for the means of Minkowski sums. In fact if K ⊂ R ,for every x ∈ R holds 1 m m X j =1 ( K − x ) = 1 m m X j =1 K − x, and taking x as the mean point of the extremals of K we reduce ourself to the same context of theLemma, which can be restated as follows. Lemma 5.2.
Let K ∈ R be a compact set such that conv ( K ) = [ a, b ] with the following property: ∃ ε > s.t. [ a, a + ε ] ∪ [ b − ε, b ] ⊂ K. Then there exist an index ℓ ∈ N depending on ε and ( b − a ) / such that ℓ ℓ X j =1 K = 12 ℓ + k k + ℓ X j =1 K for every k ∈ N .Moreover, ℓ increases with ( b − a ) / and decreases if ε increases.Proof. First we remind the reader that, as we have seen in Theorem 4.2, when we iterate M H ,after the first symmetrization we are just computing the mean12 m − = m − X j =1 M H K = M mH K. Moreover, we observe that the only difference with the previous Lemma is that we don’t have thesum with the reflection to guarantee that both the extremals are part of a set of positive measure,so we require it in the hypothesis.Now we can work with a set ˜ K := [ − M, − m ] ∪ [ m, M ] + x for a suitable x ∈ R , and the rest of the proof follows straightforward in the same way.This result permits us to show that Minkowski and Fiber symmetrizations have a certain indexof idempotency for a special class of compact sets.Consider a convex compact body L in R n and an open set set C such that its closure is included inthe interior of K . Then we say that the set K = L \ C has a convex shell . This notion generalizesthe one of convex annulus . Let us find a more operative characterization. Lemma 5.3.
Let K ∈ C n . Then K has a convex shell if and only if there exist v in the interiorof conv K and > λ > such that [ λ<ε ≤ ε∂ conv ( K − v ) ⊆ K − v. Proof. If K has a convex shell, fix ν = inf x ∈ C d ( ∂L, x ) >
0, where
C, L are the set in the definition,and take v in the interior of L =conv( K ). Then, if M = max x ∈ L k x − v k , we have that λ =( M − ν ) /M clearly satisfies the requested property.Conversely, we have that K ⊇ ( K − v ) \ λ ( K − v ) + v . The outer boundary of K is the same ofconv K , then they differ at most of an open set whose closure is contained in λ ( K − v ) + v .We will call the value ε = inf x ∈ C d ( ∂L, x ) the minimum thickness of the shell of K = L \ C .The property of owning a convex shell is stable under Minkowski and Fiber symmetrizations, aswe show in the following Lemma. Lemma 5.4. If K has a convex shell its Minkowski symmetral has a convex shell too. The sameholds for Fiber symmetrization. roof. Consider a subspace H , and observe that in general, for every convex compact body A, B in R n , ∂ ( A + B ) ⊆ ∂A + ∂B. Then, taken λ, v as in the characterization in Lemma 5.3, for every λ < ε ≤ ε∂M H (conv( K − v )) ⊆ ε ∂ conv( K − v ) + ∂R H conv( K − v )2 ⊆ ( K − v ) + R H ( K − v )2 = M H ( K − v ) . Using Lemma 4.1, M H ( K − v ) = M H K − v | H and M H (conv( K − v )) = conv( M H K − v | H ), provingour assertion.For the Fiber symmetrization, the result holds trivially working on the sections K ∩ ( H ⊥ + x ) , x ∈ H .This permits us to prove the following generalization of Lemma 5.2. Proposition 5.5.
Let K ∈ C n such that it has a convex shell. Then, for every subspace H ⊂ R n we have that there exist ℓ ∈ N dependent from the minimum thickness of the shell, the maximumwidth of conv ( M H K ) and independent from n such that M k + ℓH K = M ℓH K = M H conv ( K ) for every k ∈ N .The same result holds for Fiber symmetrization with respect to H .Proof. We start observing that, thanks to Lemma 4.1, M H K ⊆ M kH K for every k ∈ N . Moreover,as we already observed, M H K has a convex shell. Then, taking v in the interior of conv( M H K ) ∩ H ,all the intersections between M H K and the affine lines passing from v satisfy the hypothesis ofLemma 5.2, and for each one of them there exists an index ℓ u , where u ∈ S n − is the direction ofthe line, such that the corresponding intersection has idempotency degree ℓ u .Then if M := max x ∈ M H K k x − v k is the maximum ray and ε is the minimum thickness of theconvex shell, taking ℓ as the idempotency index of the set [ − M, − M + ε ] ∪ [ M − ε, M ], we havethat ℓ ≥ ℓ u for every u ∈ S n − . Now we prove that every section by affine lines from v is filledafter ℓ symmetrizations. In fact, calling s u these sections, for every k ∈ N we have the inclusions s u ⊂ M H K ⊆ M kH K = 12 ( M H K + ... + M H K | {z } k − -times ) , because M H K is H -symmetric. Then M kH K contains the mean12 k − k − X j =1 s u . Observe that this index is determined from M and ε . Then M ℓH K has a convex shell and isstar-shaped with respect to v , thus it is convex. The independence of ℓ from n is clear from theconstruction.Consider now the Fiber symmetrization with respect to H . Recalling the definition, we have thatit is the disjoint union of the Minkowski symmetrals of the sections K ∩ ( H ⊥ + x ) , x ∈ H, thus every one of them has a finite index of idempotency ℓ x , each one of them depending ona respective ray M x and thickness ε x . If we now consider the ray M and the thickness ε of F H K , obviously M ≥ M x and ε ≤ ε x for every x ∈ H . Thus, if ℓ is the corresponding index ofidempotency, ℓ ≥ ℓ x , concluding the proof.An immediate application is a generalization of Theorem 1.1 from [12]. Theorem 5.6 (Klartag) . Let < ǫ < , n > n ( ǫ ) . Given a compact set K ⊂ R n with convexshell, there exist cn log n + c ( ǫ ) n + ℓ Minkowski symmetrizations by hyperplanes that transform Kinto a body ˜ K such that (1 − ǫ ) w ( K ) B n ⊂ ˜ K ⊂ (1 + ǫ ) w ( K ) B n , where c ( ǫ ) , n ( ǫ ) are of the order of exp( cǫ − | log ǫ | ) , ℓ depends only on the thickness and maximumray of the shell and c > is a numerical constant. roof. First we consider the sequence given by the original statement of this theorem for the convexbody conv K . As we have proved before, iterating a finite number of times the same symmetrizationwe obtain a convex body. Applying the first symmetrization in this way, then we proceed with theremaining ones, and the result holds as for conv K .We conclude this paper with the proof of Theorem 1.3 preceded by a couple of technical Lemmas.This last part does not exactly require to have a convex shell. It will be sufficient to have a convexouter boundary, i.e. ∂ conv K ⊂ K, (2)and to have positive measure. Lemma 5.7.
Let K a compact set with positive measure, H a subspace. Then K is invariantunder M H if and only if | K | = | M H K | .Proof. Consider the case | K | = | M H K | . If K = M H K , thanks to Lemma 4.1 we know that K isnot convex and H -symmetric at the same time. Then, for the Brunn-Minkowski inequality, | M H K | /n = (cid:12)(cid:12)(cid:12)(cid:12) K + 12 M H K (cid:12)(cid:12)(cid:12)(cid:12) /n > | K | /n + 12 | R H K | /n = | K | /n , and the inequality is strict because K is not convex or homothetic to R H K . But this means that | M H K | > | K | , which is a contradiction.The other implication is trivial. Lemma 5.8.
Let K a compact set such that (2) holds and | K | > . Then, if its outer boundaryis H -symmetric, K is invariant under F H if and only if | K | = | F H K | .Moreover, if (2) holds, | K | > and K is invariant under Fiber symmetrization, then K is convexand H -symmetric.Proof. Having K a symmetric convex outer boundary, its outer boundary will be the same of F K ,thus if they differ from each other they do it in the inner part. Moreover, observe that for everysection the Brunn-Minkowski inequality gives (cid:12)(cid:12) F H K ∩ ( H ⊥ + x ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) K ∩ ( H ⊥ + x ) (cid:12)(cid:12) for every x ∈ H .Lemma 5.7 implies that either the two sections are equal and thus convex and H -symmetric, orthat the inequality is strict, and in general by Fubini’s Theorem | F H A | ≥ | A | for every compactset A . Then, if | K | = | F H K | , they can differ only in their sections by sets of measure zero, becausetheir outer boundary remains the same, and this is not possible being them compact.The other implication is trivial.The last assertion follows from the fact that if K is invariant under symmetrization then wehave filled all the portion of space bounded by ∂ conv K , thus K is convex, and it is obviously H -symmetric. Proof of Theorem 1.3.
We start noting that the outer boundary of K will transform as the bound-ary of conv K during the process of symmetrization. This is because in the Minkowski sum theboundary of the sum is included in the sum of the boundary, as we observed before. Moreover,Fiber symmetrization is monotone, so we have left to prove that the inner part of K will becompletely filled during the process of symmetrization.Remember that in general, if A ⊂ R n and H is a subspace, then | F H A | ≥ | A | . (3)Take H ∈ F a subset appearing infinitely often in { H m } , and let { K m j } a subsequence of { K m } whose elements are the ones preceding the symmetrization by H . Thanks to Blaschke’s selectionTheorem, there exists a further subsequence, which we call again { K m j } , converging to somecompact set ˜ K . Notice that the outer boundary of ˜ K is equal to the boundary of L . Using (3) wehave that (cid:12)(cid:12) ˜ K (cid:12)(cid:12) ≥ (cid:12)(cid:12) F H K m j (cid:12)(cid:12) ≥ (cid:12)(cid:12) K m j (cid:12)(cid:12) . m → + ∞ we obtain that (cid:12)(cid:12) ˜ K (cid:12)(cid:12) = (cid:12)(cid:12) F H ˜ K (cid:12)(cid:12) . Thanks to the generalization ofKlain’s Theorem for Fiber symmetrization of convex set, L is invariant under F H , so the shell of ˜ K is H -symmetric. Thus by Lemma 5.8 it follows that ˜ K is convex and H -symmetric. Now, ˜ K ⊂ L and they have the same boundary, thus ˜ K = L .Let { K ˜ m l } any other subsequence of { K m } . Again for (3) we have | L | ≥ | K ˜ m l | ≥ (cid:12)(cid:12) K ¯ m j (cid:12)(cid:12) , where K ¯ m j is an element of { K m j } preceding K ˜ m l in { K m } , then | L | = lim m → + ∞ | K ˜ m l | . (4)For the monotonicity of Fiber symmetrization, even if { K ˜ m l } does not converge, its outer boundarydoes, and the not convergent part will be bounded in L , which with (4) implies thatlim m → + ∞ K ˜ m l = L. Thus every subsequence of K m converges to L , which concludes the proof. In this work we have partially solved
Problem 8.4 given in [4], concerning the generalization ofKlain’s result to further symmetrizations of compact sets. Here we present a still open problem.
Problem 6.1.
Is it possible to prove a result analogue to Corollary 4.5 for the Fiber symmetrizationof general compact sets?
We have seen that an analogue of Klain’s Theorem holds for the Fiber symmetrization of compactconvex shells, and it does mainly because of the assumption of convexity for the outer boundary.Generalizing this result to general compact sets implies the challenge to control the behavior ofboundary sections of measure zero, which can change drastically the shape of the object during theprocess of symmetrization. Moreover, some families of subspaces may be more suitable if taken inaccount preexisting symmetries of the object.The approach in [3] and [4] was of a variational fashion, which is not an optimal tool when talkingabout Minkowski sum of compact sets. In [2] a different method was successful for Steiner andlater Schwarz symmetrizations, but it was based on peculiar properties of Steiner symmetrizationthat Fiber symmetrization does not possess, even though the union of the method in [2] togetherwith the one we used in Theorem 1.2 may provide in future an answer to this problem.
Aknowledgements
The author would like to thank Gabriele Bianchi and Paolo Gronchi for the insightful help andthe inputs that started this work, in particular the counterexamples in Section 3.
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