Big convex polytopes or rich hyperplanes

TL;DR

This paper investigates the minimum number of points needed in Euclidean space to guarantee either a large subset on a hyperplane or a convex configuration, providing bounds for this extremal problem.

Contribution

It establishes new upper and lower bounds for the function $ES_d(l,n)$, advancing understanding of geometric extremal configurations.

Findings

Derived bounds for $ES_d(l,n)$ in various dimensions

Identified conditions for the existence of hyperplane or convex subsets

Extended classical extremal geometric results

Abstract

For natural numbers $n$ and $l > d \geq 2$, let $ES_d(l,n)$ be the minimum $N$ such that any set of at least $N$ points in $\mathbb{R}^d$ contains either $l$ points contained in a common $(d-1)$-dimensional hyperplane or $n$ points in convex position. In this paper, we give the upper and lower bounds for $ES_d(l,n)$.