Bounding regions to plane steepest descent curves of quasi convex families

TL;DR

This paper investigates the extension of steepest descent curves for quasi convex families outside convex bodies, establishing bounding regions defined by involutes and analyzing minimal length extensions and self-contracting sets.

Contribution

It introduces a method to bound steepest descent curves using involutes of convex boundaries and characterizes minimal length extensions and conditions for self-contracting sets.

Findings

Bounding regions are defined by involutes of convex boundaries.

Extensions of minimal length are constructed within these regions.

Conditions for self-contracting sets to be subsets of steepest descent curves are established.

Abstract

Two dimensional steepest descent curves (SDC) for a quasi convex family are considered; the problem of their extensions (with constraints) outside of a convex body $K$ is studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending on $K$. These regions are bounded by arcs of involutes of the boundary of $K$ and satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self contracting sets (with opposite orientation) are considered, necessary and/or sufficients conditions for them to be subsets of a SDC are proved.