Mean value inequalities for convex and star-shaped sets

Abstract

Let \(A \subset B \subset \mathbb{R}^{n} \) and let 0 be an interior point of A. It is shown that if A and B are compact star-shaped sets at 0 and if 0 is their common barycenter, then there is a positive number κ such that for every 0 < λ ≤ κ, the set λ A is convexly majorized by B. If, in addition, B is a symmetric (i.e. −B = B) convex set, it is shown that there is a universal positive constant κn, which depends only on the dimension n, such that for every symmetric convex set A, which satisfies the relation A ⊂ κ nB and if A and B have the same barycenter, then A is convexly majorized by B.