aa r X i v : . [ m a t h . M G ] O c t A CHARACTERIZATION OF ZONOIDS
YOSSI LONKE Introduction
A function defined on the n -dimensional sphere S n − will be called zonal if its value ata point x ∈ S n − depends only on the angle between x and a fixed axis. Thus if u is a unitvector in the direction of the fixed axis, then a function is zonal with respect to u if itsvalue at x depends only on the standard inner product h x, y i . A natural way to generatezonal functions is as follows. Let f ∈ C ( S n − ), and fix a direction u ∈ S n − . Let ( O ( u ) , m )denote the subgroup of orthogonal transformations which keep the point u fixed, equippedwith the normalized Haar measure, m . Define( S u f )( x ) = Z O ( u ) f ( T x ) dm ( T )Clearly, the function S u f is zonal. Appyling this procedure to support functions leads tothe following definition. Definition 1.1.
Suppose u ∈ S n − and K is a centrally symmetric convex body, withsupport function h K . The u -spin of K is the convex body K u whose support function is S u h K .For example, computing the e n -spin of the unit cube in R n , gives:( S e n h B ∞ n )( x ) = Z O ( e n ) k T x k dm ( T ) = c n n X i =1 x i ! / + | x n | , ( c n is a positive number depending on n ), which is the support function of a cylinder. Thisillustrates the choice of the word ’spin’.Thus, with each body there is associated a system of rotation-bodies, one for eachpossible direction. Since S u f ( u ) = f ( u ) for each u ∈ S n − and every function f , a body isseen to be uniquely defined by its spins. The main result of this note is the following. Theorem 1.2.
A centrally symmetric convex body is a zonoid if and only if all its spinsare zonoids.
Date : February 2020. Preliminaries C ∞ ( S n − ) is the Frechet space of functions on S n − that have derivatives of every order.Elements of this space are called test functions . Its dual space, denoted D ( S n − ), is thespace of distributions on S n − . If a subscript e is added to any of the above spaces, it is todesignate the subspace of even objects.The cosine transform is the operator C : C ∞ e ( S n − ) → C ∞ e ( S n − ) defined by:( C f )( x ) = Z S n − |h x, y i| f ( y ) dσ n − ( y ) , where σ n − is the normalized rotation-invariant measure on the sphere. It is well knownthat C is a continuous bijection of C ∞ e ( S n − ) onto itself, and that it can be extended byduality to a bi-continuous bijection of the dual space D e ( S n − ). Hence, if ρ is a distributionand f ∈ C ∞ e ( S n − ) is a test function, then hC ρ, f i = h ρ, C f i Since C ∞ e ( S n − ) and its dual space, the even measures, are both naturally embeddedin D e ( S n − ), it makes sense to speak about the cosine transform of a measure, or ofa continuous function. A fundamental connection between distributions and centrallysymmetric convex bodies was discovered by Weil, in [1]. Weil proved that for every centrallysymmetric convex body K ⊂ R n there corresponds a unique distribution ρ K (called the generating distribution of K ) such that C ρ K = h K , where h K is the support function of K .Suppose f ∈ C ∞ ( S n − ). Then for every direction u ∈ S n − , the function S u f also belongsto C ∞ ( S n − ). Therefore, S u can be defined to act on distributions by duality: h S u ρ, f i = h ρ, S u f i , ρ ∈ D ( S n − ) , f ∈ C ∞ ( S n − )A routine verification shows that the transforms S u and C commute on test functions, andtherefore also as transforms of distributions.3. Proof of Theorem 1.1 If K is a zonoid, then h K = C µ for some positive measure µ , and for every u ∈ S n − , S u h K = S u ( C ( µ )) = C ( S u µ )Since S u µ is a positive measure for every µ , every spin of K is a zonoid. This proves theeasy part of the theorem.Suppose K is a centrally symmetric convex body every spin of which is a zonoid. That is,for each direction u ∈ S n − there exists a positive measure µ u such that S u h K = C µ u . Thereexists a distribution ρ such that h K = C ρ , and so the assumption reduces to C ( S u ρ ) = C µ u ,where the commuting of S u and C was used. Since C is one-to-one, the distribution ρ isseen to satisfy S u ρ = µ u for every u ∈ S n − . It therefore remains to prove: Lemma 3.1.
A distribution is positive if and only if S u ρ is positive for every u ∈ S n − . CHARACTERIZATION OF ZONOIDS 3
Proof.