We prove that the Euclidean ball is the unique convex body with the properly that all its sections through a fixed point are convex bodies of constant width. Furthermore, we characterize those convex bodies which are sections of convex bodies of constant width.
1. INTRODUCTION AND NOTATION
The main purpose of this paper is to prove the following theorem: THEOREM 1. Let 2 <~ k < n, let K c E_" be an n-dimensional convex body and let Po be a point of F_ n with the property that every k-section of K through Po is a convex body of constant width. Then K is a Euclidean n-ball.
When we say that every k-section of K through Po is a convex body of constant width we mean that if H is a k-plane of fl:n through Po, then 17 c~ K is either empty, a single point or a k-dimensional convex body of constant width.
Under differentiability conditions on K and its sections, Theorem 1 was proved by Siiss [2], when n = 3 and Poe int K.
A k-dimensional convex body K c E k will be called R-convex, if there is a sufficiently large number r > 0 and for every p e bdr K there is a Euclidean kball B c E k of diameter r which contains K and has p on its boundary. By Theorem 1 we know that not every k-section of a convex body of constant width, different from a Euclidean ball, has constant width, thus it would be interesting to know which convex bodies are k-sections of convex bodies of constant width. In this direction we shall prove that a k-dimensional convex body K is the k-section of an n-dimensional convex body of constant width if and only if K is R-convex.
In this paper E n will denote Euclidean n-space, where it is always assumed that n ~> 2. A Euclidean n-ball in ~:n, or simply a Euclidean ball in iF", will be a subset of E" homothetic to {x e ~:" I Ilxll ~< 1}. If a, b e iF", then [a, b] and (a, b) will denote the closed and open intervals with extreme points a and b, respectively. If K is a (k + 1)-dimensional convex body and [a, b] is a chord of * Research supported by the Alexander von Humboldt-Foundation.