Large convexly independent subsets of Minkowski sums

TL;DR

This paper investigates the maximum size of convexly independent subsets in Minkowski sums and related geometric configurations, providing new bounds and asymptotic results in various dimensions.

Contribution

It establishes new lower bounds for convexly independent pairs in Minkowski sums in 2D and bounds in 3D, and relates these to maximum nonparallel unit distance pairs in convex normed spaces.

Findings

Proves $E_2(n) ext{ is at least } ext{Omega}(n\sqrt{ ext{log} n})$.

Bounds $E_3(n)$ between $ ext{floor}(n^2/3)$ and $(3/8)n^2 + O(n^{3/2})$.

Shows $W_2(n)$ is asymptotically equivalent to $E_2(n)$ and provides asymptotics for $W_d(n)$ in higher dimensions.

Abstract

Let $E_d(n)$ be the maximum number of pairs that can be selected from a set of $n$ points in $R^d$ such that the midpoints of these pairs are convexly independent. We show that $E_2(n)\geq \Omega(n\sqrt{\log n})$, which answers a question of Eisenbrand, Pach, Rothvo\ss, and Sopher (2008) on large convexly independent subsets in Minkowski sums of finite planar sets, as well as a question of Halman, Onn, and Rothblum (2007). We also show that $\lfloor\frac{1}{3}n^2\rfloor\leq E_3(n)\leq \frac{3}{8}n^2+O(n^{3/2})$. Let $W_d(n)$ be the maximum number of pairwise nonparallel unit distance pairs in a set of $n$ points in some $d$-dimensional strictly convex normed space. We show that $W_2(n)=\Theta(E_2(n))$ and for $d\geq 3$ that $W_d(n)\sim\frac12\left(1-\frac{1}{a(d)}\right)n^2$, where $a(d)\in N$ is related to strictly antipodal families. In fact we show that the same asymptotics hold without the requirement that the unit distance pairs form pairwise nonparallel segments, and also if diameter pairs are considered instead of unit distance pairs.