Analogues of the central point theorem for families with $d$-intersection property in $\mathbb R^d$

TL;DR

This paper explores extensions of the central point and Tverberg's theorems for families of convex sets in ^d with the property that any subfamily of size at most d intersects, providing new geometric insights.

Contribution

It establishes analogues of classical theorems for convex families with the d-intersection property, broadening their applicability in geometric combinatorics.

Findings

Proved new versions of the central point theorem for these families.

Extended Tverberg's theorem to families with the d-intersection property.

Provided geometric conditions under which these theorems hold.

Abstract

In this paper we consider families of compact convex sets in $\mathbb R^d$ such that any subfamily of size at most $d$ has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg's theorem for such families.