Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order

TL;DR

This paper proves that uniformly convex sets in Hilbert spaces with a second-order modulus of convexity can be represented as intersections of fixed-radius closed balls, providing an estimate for this radius.

Contribution

It establishes a new characterization of uniformly convex sets with second-order modulus in Hilbert spaces as intersections of fixed-radius balls, including an estimate of the radius.

Findings

Uniformly convex sets with second-order modulus are intersections of fixed-radius balls.

An explicit estimate of the radius of these balls is provided.

The result enhances understanding of the geometric structure of convex sets in Hilbert spaces.

Abstract

We prove that in the Hilbert space every uniformly convex set with modulus of convexity of the second order at zero is an intersection of closed balls of fixed radius. We also obtain an estimate of this radius.