Tangent metric spaces to starlike sets on the plane

TL;DR

This paper characterizes tangent spaces to starlike sets in the plane, showing they are isometric to the minimal closed cone containing the set, and discusses tangent spaces to convex sets.

Contribution

It provides a precise description of tangent spaces to starlike sets and offers a partial converse, advancing geometric understanding of these structures.

Findings

Tangent spaces to starlike sets are isometric to the minimal closed cone containing the set.

A partial converse to the main theorem is established.

Tangent spaces to convex sets are also analyzed.

Abstract

Let $A\subseteq\mathbb C$ be a starlike set with a center $a$. We prove that every tangent space to $A$ at the point $a$ is isometric to the smallest closed cone, with the vertex $a$, which includes $A$. A partial converse to this result is obtained. The tangent space to convex sets is also discussed.