Preservation of F-convexity under the heat flow

TL;DR

This paper introduces F-convexity as a generalization of power convexity, characterizes which F-convexities are preserved under heat flow in Euclidean space and convex domains, and identifies the extremal cases.

Contribution

It extends the concept of convexity to F-convexity and characterizes its preservation under heat flow in various settings, including Euclidean space and convex domains.

Findings

Characterization of F-convexities preserved under heat flow

Identification of strongest and weakest preserved F-convexities

Extension of convexity preservation results to Dirichlet heat flow

Abstract

We introduce the notion of F-convexity as a general extension of power convexity. We characterize the F-convexities preserved under the heat flow in the n-dimensional Euclidean space, and identify the strongest and the weakest ones among them. We also characterize the F-convexities preserved under the Dirichlet heat flow in convex domains.