# Convex Sets and Minimal Sublinear Functions

## TL;DR

This paper characterizes the support function of a specific convex set related to a given convex set with the origin in its interior, showing it is the minimal sublinear function defining that set.

## Contribution

It establishes that the support function of a particular convex set is the smallest among all sublinear functions representing the original convex set.

## Key findings

- Support function is the minimal among sublinear functions for the set
- Characterization of convex sets via sublinear functions
- Theoretical insight into convex analysis and support functions

## Abstract

We show that, given a closed convex set $K$ containing the origin in its interior, the support function of the set $\{y\in K^*: \exists x\in K\mbox{ such that } \langle x,y \rangle =1\}$ is the pointwise smallest among all sublinear functions $\sigma$ such that $K=\{x: \sigma(x)\leq 1\}$.