Stable determination of convex bodies from sections

Abstract

A star body (with respect to the origin 0) in ℝ d ( d ≧ 3) which has 0 as center of symmetry is uniquely determined by the ( d − 1)-dimensional volumes of its sections with hyperplanes through 0. Without the symmetry assumption, we show that a star body is uniquely determined by the volumes and centroids of its hyperplane sections through 0. For convex bodies, we prove a stability version of this result.