Hypomonotonicity of the normal cone and proximal smoothness

TL;DR

This paper characterizes proximally smooth sets via the monotonicity of their normal cones in Banach spaces, providing bounds related to the space's smoothness and convexity.

Contribution

It offers a complete characterization of proximally smooth sets through normal cone properties in general Banach spaces, with explicit bounds based on space moduli.

Findings

Complete characterization of proximally smooth sets via normal cone monotonicity

Exact bounds for monotonicity inequality in terms of Banach space moduli

Application to arbitrary uniformly convex and smooth Banach spaces

Abstract

In this paper we study the properties of the normal cone to the proximally smooth set. We give the complete characterization of the proximally smooth set through the monotony properties of its normal cone in an arbitrary uniformly convex and uniformly smooth Banach space. We give the exact bounds for right-hand side in the monotonicity inequality for normal cone in terms of the moduli of smoothness and convexity of a Banach space.