Intuitive norms are Euclidean

TL;DR

The paper defines intuitive norms as those for which the geometric median $GM_K(W)$ of any finite point set $P$ with weights $W$ lies in the convex hull $\mathrm{conv}(P)$. It shows that this property holds for all norms in the plane, and that in dimensions $n\ge3$ it forces the norm to be Euclidean, i.e., the unit ball must be an ellipsoid. The analysis combines sub-differential techniques, convex-separation arguments in the plane, and a 3D ellipsoid characterization (via a smooth-case and a density-continuity extension) to establish a rigidity result: only ellipsoidal norms are intuitive in higher dimensions. This yields a sharp dichotomy between planar universality and higher-dimensional rigidity, with implications for geometric median behavior under general 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.

Figures (1)

• Figure 1: A two-dimensional symmetric convex body $K$. The points $P$ are in the upper half plane. The two marked points are $x_*,-x_*$. The dotted tangent line is in direction $u$.

Theorems & Definitions (27)

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• Claim 4