Intuitive norms are Euclidean

TL;DR

This paper characterizes all Euclidean norms on r^n that satisfy a geometric median property, revealing a unique class of norms with this intuitive convex hull inclusion feature.

Contribution

It provides a complete characterization of intuitive norms, a new class of norms defined by a median convex hull property.

Findings

Euclidean norms are the only intuitive norms.

The median convex hull property uniquely characterizes Euclidean norms.

The paper offers a geometric criterion for identifying Euclidean norms.

Abstract

We call a norm on $\mathbb{R}^n$ intuitive if for every points $p_1,\ldots,p_m$ in $\mathbb{R}^n$, one of the geometric medians of the points over the norm is in their convex hull. We characterize all intuitive norms.