Convex Bodies with affinely equivalent projections and affine bodies of revolution
aa r X i v : . [ m a t h . M G ] M a y CONVEX BODIES WITH AFFINELY EQUIVALENT PROJECTIONSAND AFFINE BODIES OF REVOLUTION
LUIS MONTEJANO
Abstract.
In this paper, we study affine bodies of revolution. This will allow us toprove that a convex body all whose orthogonal n -projections are affinely equivalent isan ellipsoid, provided n ≡ , , n >
1, with the possible exemption of n = 133.Our proof uses convex geometry and topology of compact Lie groups. Introduction
The purpose of this paper is to prove the following Theorem.
Theorem 1.1.
Suppose B ⊂ R N is a convex body, all of whose orthogonal projections onto n -dimensional linear subspaces, for some fixed integer n , < n < N , are affinely equivalent.If n ≡ , , mod , then B is an ellipsoid, with the possible exemption of n = 133 . The analogue for sections is the following statement which is equivalent to the real Banachconjecture [1] stated en 1932.
Suppose B ⊂ R N is a convex body containing the origin in its interior, all of whosesections through n -dimensional linear subspaces, for some fixed integer n , < n < N , areaffinely equivalent. Then B is an ellipsoid , Real Banach Isometric Conjecture
Suppose V be a real Banach N -dimensional spaceall of whose n -dimensional subspaces, for some fixed integer n , < n < N , are isometricallyisomorphic to each other. Then V is a Hilbert space .The conjecture was proved by Gromow [4] in 1967, for n = even or N > n + 1 and byMontejano et.al. [2], for n ≡ n = 133. The historybehind this conjecture can be read in [11]. It is also worthwhile to see [10] and the notes ofSection 9 of [8]. Furthermore, for more about bodies having affinely equivalent or congruentsections or projections, see Problem 3.3, Note 3.2 and Problem 7.4, Note 7.2 of RichardGardner’s Book [3].The reason for the strange exception n = 133 is because 133 is the dimension of theexceptional Lie group E for which it was not possible to prove Theorem 1.6 of [2], crucialfor the proof of our Theorem 2.4.The proof of Theorem 1.1 combines two main ingredients: convex geometry and topol-ogy and geometry of transformation groups. The first part of the article consists of usingtopological methods to show that, under the hypotheses of Theorem 1.1, all orthogonal pro-jections of B are affinely equivalent to a symmetric body of revolution. Section 3 is devotedto prove the following characterization of ellipsoids. Date : Preliminary Version.
Theorem 1.2.
A symmetric convex body B ⊂ C n +1 , n ≥ , all of whose orthogonal projec-tions onto hyperplanes are linearly equivalent to a fixed body of revolution, is an ellipsoid. Reducing the structure group of the tangent bundle of the sphere
During this section, we assume B ⊂ R n +1 is a convex body, all of whose orthogonalprojections onto n -dimensional linear subspaces are lineally equivalent, n ≡ , , n > n = 133. If this is the case, we shall prove that there is a symmetric convex bodyof revolution K ⊂ R n with the property that every orthogonal projection of B is linearlyequivalent to K .The link to topology is via a beautiful idea that traces back to the work of Gromov [4].It consists of the following key observation. Lemma 2.1.
Let B ⊂ R n +1 be a symmetric convex body, all of whose orthogonal projectionsonto hyperplanes are linearly equivalent to some fixed symmetric convex body K ⊂ R n . Let G K := { g ∈ SO ( n ) | g ( K ) = K } be the group of symmetries of K . Then the structure groupof the tangent bundle of S n can be reduced to G K .Proof. Consider the symmetric ellipsoid E of minimal volume containing B ∩ R n . For theexistence of L¨owner-Johns minimal ellipsoids, see Gruber [5]. By translation and dilatationof the principal axes of this ellipsoid, we obtain an affine isomorphism f : R n → R n suchthat f ( E ) is the unit ball of R n . Let K = f ( B ∩ R n ). Obviously, the structure group of thetangent bundle of S n can be reduced to { g ∈ GL n ( R ) | g ( K ) = K } . See Section 3.1 of [2]for the completely analogous proof of Lemma 1.5, for sections instead of projections, as wellas a brief reminder about structure groups of differentiable manifolds and their reductions.Obviously, K is affinely equivalent to every orthogonal projection of B . The next step is toobserve that { g ∈ GL n ( R ) | g ( K ) = K } is a subgroup of the orthogonal group O ( n ). This isso because every linear homeomorphism that fixes K also fixes the unit ball. Furthermore,if the structure group of the tangent bundle of S n can be reduced to { g ∈ O ( n ) | g ( K ) = K } ,then it can be reduced to G K . (cid:3) Lemma 2.1 can be also interpreted through the notion of fields of convex bodies tangentto S n as follows. See, for example, Mani [7] and Montejano [9]. For every u ∈ S n , let u ⊥ be the hyperplane subspace orthogonal to u and F ( u ) be the unique n -dimensional ellipsoidof least volume containing the orthogonal projection π u ( B ) of B in the direction u . Theaffine transformation β u which maps F ( u ) onto the n -dimensional Euclidean unit ball in u ⊥ by translating and dilating its principal axes is a continuous function of u . Then all β u ( π u ( B )), for u ∈ S n are congruent, so there is a complete turning of β e ( π ( B )), where π is the orthogonal projection onto R n and e = (0 , . . . , , β e ( π ( B )),is centrally symmetric, and so is any other orthogonal projection of B . This implies that Corollary 2.2.
Let B ⊂ R N be a convex body, all of whose orthogonal projections onto n -dimensional subspaces are affinely equivalent, < n < N . Then B is centrally symmetric.Proof. If n + 1 = N , the proof follows immediately from the above paragraph and Lemma2 of [9]. The corollary follows by induction using the obvious fact that every orthogonalprojection of a convex body is centrally symmetric if and only if this body is centrallysymmetric. (cid:3) As an immediate consequence of Aleksandrov Theorem (Theorem 2.11.1 of [8]), we have
Theorem 2.3.
Let B ⊂ R N be a symmetric convex body, all of whose orthogonal projectionsonto n -dimensional subspaces are congruent, < n < N . Then B is a ball. ONVEX BODIES WITH AFFINELY EQUIVALENT PROJECTIONS AND AFFINE BODIES OF REVOLUTION3
Let us prove now the main result of this section
Theorem 2.4.
Suppose B ⊂ R n +1 is a convex body, all of whose orthogonal projectionsonto hyperplanes, for some fixed integer n , < n < N , are affinely equivalent. If n ≡ , , mod and n = 133 , then there exist a symmetric body of revolution K such that everyorthogonal projection of B onto a hyperplane subspace is linearly equivalent to K .Proof. By the first part of this section, we now that there exist a symmetric convex body K such that every orthogonal projection of B onto a hyperplane is linearly equivalent to K and such that the structure group of S n can be reduced to G K = { g ∈ SO ( n ) | g ( K ) = K } .If n is even and n = 6 , Theorem 1A) of Leonard [6] implies that G = SO ( n ) and therefore K is a ball. In particular, K is a body of revolution. If n = 6, then Theorem 1A) of Leonard[6] implies that G = SU (3) or U (3), but in both cases the action in S is transitive, whichimplies again that K is a ball. If n ≡ n = 133, then Theorem 1.6 of [2] impliesthat K is a body of revolution. (cid:3) Affine Bodies of Revolution A symmetric convex body is a compact convex subset of a finite dimensional real vectorspace with a nonempty interior, invariant under x
7→ − x. A hyperplane is a codimension1 linear subspace. An affine hyperplane is the translation of a hyperplane by some vector.Two sets, each a subset of a vector space, are linearly (respectively, affinely ) equivalent ifthey can be mapped to each other by a linear (respectively, affine) isomorphism betweentheir ambient vector spaces. Given an affine k -dimensional plane Λ in R n , we denote by Λ ⊥ the corresponding ( n − k )-dimensional subspace orthogonal to Λ.An ellipsoid is a subset of a vector space which is affinely equivalent to the unit ball ineuclidean space. A convex body K ⊂ R n is a body of revolution if it admits an axis ofrevolution , i.e., a 1-dimensional line L such that each section of K by an affine hyperplane∆ orthogonal to L is an n − A ∩ L (possiblyempty or just a point). If L is an axis of revolution of K then, L ⊥ , is the associated hyperplane of revolution . An affine body of revolution is a convex body affinely equivalent toa body of revolution. The images, under an affine equivalence, of an axis of revolution andits associated hyperplane of revolution of the body of revolution are an axis of revolutionand associated hyperplane of revolution of the affine body of revolution (not necessarilyperpendicular anymore). Clearly, an ellipsoid centered at the origin is an affine symmetricbody of revolution and any hyperplane serves as a hyperplane of revolution.The aim of this section is to prove the following theorem. Theorem 3.1.
Let B ⊂ R n +1 , n ≥ , be a symmetric convex body, all of whose orthogonalprojections are symmetric affine bodies of revolution. Then, at least one of its projectionsis an ellipsoid. The first step is to prove that the projection of an affine body of revolution is an affinebody of revolution. For that purpose, we need to prove first the following technical lemma.
Lemma 3.2.
Let Γ ⊂ R n , n ≥ , be an affine hyperplane and let C be an ( n − -dimensionalball contained in Γ with center at x . Let H be a hyperplane subspace non parallel to H andlet π : R n → H be the orthogonal projection. Then, π ( C ) ⊂ H is an ellipsoid of revolutionwith axis of revolution the line L , where L = ( x + (Γ ⊥ + H ⊥ )) ∩ H . L. MONTEJANO
Proof.
Suppose without loss of generality the center x of the ( n − C is the origin.Consider the linear isomorphism f : Γ → H given by f ( z )) = π ( z ). Consequently, π ( C ) = f ( C ) is an ellipsoid. Note that f is the identity in the hyperplane subspace Γ ∩ H of H .Therefore, π ( C ) = f ( C ) is an ellipsoid of the revolution with axis (Γ ⊥ + H ⊥ ) ∩ H , as wewished. (cid:3) Lemma 3.3.
Let K ⊂ R n , n ≥ , be an affine body of revolution with axis of revolution theline L and let π be an orthogonal projection along the -dimensional subspace ℓ . Then π ( K ) is an affine body of revolution. Moreover, if L is parallel to ℓ , then π ( K ) is an ellipsoid andif not, π ( L ) is an axis of revolution of π ( K ) .Proof. Let us first prove the case in which K is a body of revolution. We wall prove thatin this case, if L is parallel to ℓ , then π ( K ) is ball and if not, π ( K ) is a body of revolutionwith axis π ( L ).Suppose L is not parallel to ℓ . Let P = ℓ + L be the 2-dimensional subspace generatedby ℓ and L . For every x ∈ L , let C x = ( x + L ⊥ ) ∩ K , then C x is either empty, the point x or a ball contained in ( x + L ⊥ ) and center at x . Therefore K = ∪ { x ∈ L } C x , and(1) π ( K ) = ∪ { x ∈ L } π ( C x ) . By Lemma 3.2, π ( C x ) is an ellipsoid of revolution with axis π ( L ). Consequently, by(1), π ( K ) is the union of ellipsoids of revolution, all to them with the same axis π ( L ).Consequently, π ( K ) is a body of revolution with axis π ( L ).For the proof of the general case of the lemma we may assume, after a linear isomorphism,that K is a body of revolution and π : R n → H is an affine projection in the direction of ℓ onto the hyperplane subspace H (not necessarily orthogonal to ℓ ). Suppose L is not parallelto L .Consider K ⊕ ℓ = { x ∈ R n | ( x + ℓ ) ∩ K = ∅} . By definition π ( K ) = ( K ⊕ ℓ ) ∩ H and by the first part of the proof, K = ( K ⊕ ℓ ) ∩ ℓ ⊥ is a body of revolution with axis L = ( L + ℓ ) ∩ ℓ ⊥ .Let f : ℓ ⊥ → H be the linear map given by f ( z ) = ( z + ℓ ) ∩ H , for every z ∈ ℓ ⊥ . Then f ( K ) = π ( K ) and f ( L ) = π ( L ). Consequently π ( K ) is an affine body of revolution withaxis of revolution π ( L ), as we wished. (cid:3) Lemma 3.4.
Let K ⊂ R n , n ≥ , be an affine body of revolution with hyperplane ofrevolution H and let π be an orthogonal projection along the line ℓ . Suppose that ℓ isparallel to H and π ( K ) is an ellipsoid. Then K is an ellipsoid.Proof. Since K is an affine body of revolution with hyperplane of revolution H and ℓ ⊂ H ,the shadow boundary S∂ ( K, ℓ ) = { x ∈ bd K ∩ L ′ | L ′ is a tangent line of K parallel to ℓ } has the following property: there is a hyperplane Γ such that L ⊂ Γ and Γ ∩ bd K = S∂ ( K, ℓ ).This implies that π (Γ ∩ K ) = π ( K ). Consequently, the section Γ ∩ K is an ellipsoid and byLemma 2.5 of [2], K is an ellipsoid. (cid:3) From now on, let B ⊂ R n +1 , n ≥
4, be a symmetric convex body, all of whose orthogonalprojections onto hyperplanes are non-elliptical, affine bodies of revolution.
ONVEX BODIES WITH AFFINELY EQUIVALENT PROJECTIONS AND AFFINE BODIES OF REVOLUTION5
Remember that for every line ℓ ⊂ R n +1 , we denote by ℓ ⊥ the hyperplane subspace of R n +1 orthogonal to ℓ . Furthermore, by Lemma 2.3 of [2], denote by L ℓ the unique axisof revolution of the orthogonal projection of B onto ℓ ⊥ , by H ℓ the corresponding ( n − B onto ℓ ⊥ and finallydenote by N ℓ ⊂ ℓ ⊥ the 1-dimensional subspace orthogonal to H ℓ . That is, N ℓ = H ⊥ ℓ ∩ ℓ ⊥ .Note that by the symmetry, both L ℓ and H ℓ contain the origin.We claim that the assignations ℓ → L ℓ and ℓ → N ℓ are continuos functions of ℓ . Theproof is completely analogous to the proof of Lemma 2.7 of [2]. Lemma 3.5.
Let B ⊂ R n +1 , n ≥ , be a symmetric convex body, all of whose hyperplaneprojections are non-elliptical, affine bodies of revolution. Let P be the plane generated bytwo different lines ℓ and ℓ and suppose N ℓ ⊂ ℓ ⊥ , then Π P ( L ℓ )) = Π P ( L ℓ ) , where Π P is the orthogonal projection along P .Proof. We shall first prove that ℓ ⊥ ∩ ℓ ⊥ ∩ B is a non-elliptical affine body of revolution. Forthat purpose, first note that ℓ ⊥ ∩ ℓ ⊥ is orthogonal to P . Furthermore, N ℓ ⊂ ℓ ⊥ impliesthat N ℓ ⊂ ℓ ⊥ ∩ ℓ ⊥ and hence by Lemma 3.4, if ℓ ⊥ ∩ ℓ ⊥ ∩ B is an ellipsoid, then so is ℓ ⊥ ∩ B ,contradicting our hypothesis. If this is the case, by Lemma 2.3 of [2], since n − ≥
3, weconclude that ℓ ⊥ ∩ ℓ ⊥ ∩ B has only one axis of revolution ∆. Thus, by Lemma 3.3, consideringthe orthogonal projection of ℓ ⊥ onto ℓ ⊥ ∩ ℓ ⊥ along P ∩ ℓ ⊥ , we have that ∆ = Π P ( L ℓ ), buton the other hand, considering the orthogonal projection of ℓ ⊥ onto ℓ ⊥ ∩ ℓ ⊥ along P ∩ ℓ ⊥ ,we have that ∆ = Π P ( L ℓ ). (cid:3) Before giving the proof of Theorem 3.1, we need a technical topological lemma
Lemma 3.6.
Let φ, ψ : RP n → RP n two continuos maps, both homotopic to the identity, n ≥ . Then φ − ( RP n − ) ∩ ψ − ( RP n − ) = ∅ .Proof. The closed set φ − ( RP n − ) can be thought as the carrier of σ ∈ H ( RP n ), where σ ∈ H ( RP n ) is the generator of the cohomology ring H ∗ ( RP n ). Similarly, ψ − ( RP n − ) canbe thought as the carrier of σ ∈ H ( RP n ). Since n ≥ σ is not zero and consequently φ − ( RP n − ) ∩ ψ − ( RP n − ) = ∅ . (cid:3) Proof of Theorem 3.1
Suppose not, suppose that B is a symmetric convex body allof whose orthogonal projection onto hyperplanes sections are non-elliptical, affine bodies ofrevolution. For every line ℓ ⊂ R n +1 , let H ℓ , L ℓ and N ℓ as in the paragraphs before Lemma3.5.Given the line ℓ ⊂ R n +1 , our next purpose is to show that there exist a line ℓ ⊂ R n +1 such that(1) N ℓ ⊂ ℓ ⊥ ∩ L ⊥ ℓ , and(2) L ℓ ⊂ ℓ ⊥ ∩ L ⊥ ℓ . For that purpose, note that the continuos assignations ℓ → L ℓ and ℓ → N ℓ can bethought as continuos maps from RP n into itself, moreover, since ℓ is orthogonal to L ℓ andalso orthogonal to N ℓ , the assignations ℓ → L ℓ and ℓ → N ℓ can be thought as continuousmaps from RP n into itself, which are homotopic to the identity. By Lemma 3.6, there exista line ℓ ⊂ R n +1 such that N ℓ ⊂ ℓ ⊥ ∩ L ⊥ ℓ and L ℓ ⊂ ℓ ⊥ ∩ L ⊥ ℓ . Note now that both L ℓ and N ℓ are contained in ℓ ⊥ ∩ ℓ ⊥ . If P is the plane orthogonalto ℓ ⊥ ∩ ℓ ⊥ , then N ℓ is orthogonal to P and thus P ∩ ℓ ⊥ ⊂ H ℓ . By Lemma 3.5, Π P ( L ℓ )) =Π P ( L ℓ ), where Π P is the orthogonal projection along P . L. MONTEJANO
On the other hand, L ℓ ⊂ ℓ ⊥ ∩ ℓ ⊥ implies that Π P ( L ℓ )) = L ℓ = Π P ( L ℓ ). Finally, since L ℓ is orthogonal to L ℓ in ℓ ⊥ , then Π P ( L ℓ ) is orthogonal to L ℓ , contradicting the factthat L ℓ = Π P ( L ℓ ). (cid:3) Remark 3.7.
We know that the projection or section of a convex body of revolution B isagain a convex body of revolution. The converse of this result, as far as we know, is an openproblem. Let us state a somewhat more precise question: Let B ⊂ R n +1 , n ≥ , be a convex body containing the origin in its interior. Supposeevery hyperplane section of B (projection onto a hyperplane) is a body of revolution, is B necessarily a body of revolution? Note that our Theorem 3.1 points in that direction4.
The proof of Theorem 1.1
The proof of Theorem 1.2 follows immediately from Theorem 3.1, because by Theorem2.12.5 of [8], a convex body all whose hyperplane projections are ellipsoids is an ellipsoid.The codimension 1 case of Theorem 1.1 follows directly from Corollary 2.2, Theorem 2.4and Theorem 1.2. The rest of the proof follows from the fact that a convex body all whosehyperplane projections are ellipsoids is an ellipsoid.
Acknowledgments.
Luis Montejano acknowledges support from CONACyT under project166306 and support from PAPIIT-UNAM under project IN112614.
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