Detecting Symmetry in Star Bodies

Let K be a convex body in R, i.e. a compact convex set with a non-empty interior. We say that K is origin-symmetric if K = −K. The presence of origin-symmetry is an essential assumption in various problems. Many results that hold for origin-symmetric convex bodies fail in the absence of the symmetry condition. For example, origin-symmetric convex bodies are uniquely determined by the volumes of their projections or central sections, while this is not true for general convex bodies; see [G]. Thus, detecting symmetry in convex bodies is one of the fundamental questions in convex geometry and geometric tomography. For some results in this direction the reader is referred to [AMM], [F], [Gro1], [Gro2], [Gru], [HS], [MM], [MMO], [Sch]; see also [RYZ] for open problems. In this paper we suggest a new method of detecting symmetry. Let K be a star body and let C(ξ, z) be the cone {x ∈ R : x · ξ = |x|z}, where ξ ∈ S, z ∈ (−1, 1), and x · ξ = x1ξ1 + x2ξ2 + ... + xnξn is the usual inner product in R. In this notation, z is the cosine of the angle between x and ξ. For z ∈ (−1, 1), we define the conical section function CK,ξ(z) by