Local geometry of feasible regions via smooth paths

TL;DR

This paper introduces a new approximate convexity property for feasible regions defined by smooth maps, revealing their local geometry and path-connectedness, which generalizes known properties for convex sets and smooth manifolds.

Contribution

The paper defines a novel approximate convexity property applicable to feasible regions from maps that are once differentiable, extending geometric understanding beyond convex sets and smooth manifolds.

Findings

Feasible regions with the new property allow smooth paths that are nearly straight.

Such regions are locally normally embedded in Euclidean space.

The property applies to prox-regular sets as well.

Abstract

Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case of feasible regions consisting of pre-images of convex sets under maps that are once (but not necessarily twice) continuously differentiable, the geometry is less transparent. We define a new approximate convexity property, that holds both for such feasible regions and also for all prox-regular sets. This new property requires that nearby points can always be joined by smooth feasible paths that are almost straight. In particular, in the terminology of real algebraic geometry, such feasible regions are locally normally embedded in the Euclidean space.