A generalisation of Sylvester's problem to higher dimensions

TL;DR

This paper generalizes Sylvester's problem to higher dimensions by defining a function that measures the minimal number of hyperplanes intersecting a set of points in a specific way, extending classical geometric concepts.

Contribution

It introduces the function e_d(n) to quantify hyperplane intersections in higher-dimensional point sets, generalizing Sylvester's problem beyond two dimensions.

Findings

Defines the function e_d(n) for higher-dimensional sets

Establishes bounds or properties of e_d(n)

Extends classical Sylvester's problem to d-dimensional space

Abstract

In this article we consider $S$ to be a set of points in $d$-space with the property that any $d$ points of $S$ span a hyperplane and not all the points of $S$ are contained in a hyperplane. The aim of this article is to introduce the function $e_d(n)$, which denotes the minimal number of hyperplanes meeting $S$ in precisely $d$ points, minimising over all such sets of points $S$ with $|S|=n$.