A sharp $p$-subadditive bound for the $l_p$ Hausdorff distance from convex hull

TL;DR

This paper establishes a sharp subadditive bound for the $l_p$ Hausdorff distance from convex hulls in two dimensions, revealing a precise relationship with Minkowski summation.

Contribution

It proves a sharp subadditivity property of the $l_p$ Hausdorff distance in 2D, with a specific multiplicative factor, advancing understanding of geometric bounds.

Findings

$(d^{(l_p)})^p$ is subadditive up to a factor of $ ext{max}\{1,2^{p-2} ight",

The bound is shown to be sharp in the case $n=2$ and $1 \\leq p < \\infty$.

Abstract

We study the $l_p$ Hausdorff distance from convex hull of a compact set $A\subset\mathbb{R}^n$, which is the distance \begin{equation*} d^{(l_p)}(A):=\sup_{x\in conv(A)}\inf_{a\in A}\|x-a\|_p, \end{equation*} where $\|\cdot\|_p$ is the $l_p$-norm on $\mathbb{R}^n$. We prove that when $n=2$ and $1\leq p<\infty$, the function $(d^{(l_p)})^p$ is subadditive with respect to Minkowski summation, up to multiplication by the factor $\max\{1,2^{p-2}\}$, and we observe that this bound is sharp.