Some generalizations on affine invariant points

TL;DR

This paper extends Grünbaum's conjecture on affine invariant points to a broader topological context and demonstrates its validity for various convex set families under similarity transformations.

Contribution

It provides a more general topological version of Grünbaum's conjecture and applies it to different convex set families beyond convex bodies.

Findings

Proved a topological generalization of Grünbaum's conjecture.

Showed the conjecture holds for convex sets under similarity group actions.

Extended the validity of the conjecture to non-convex families.

Abstract

In this note we prove a more general (and topological) version of Gr\"unbaum's conjecture about affine invariant points. As an application of our result we show that, if we consider the action of the group of similarities, Gr\"unbaum's conjecture remains valid in other families of convex sets (not necessarily convex bodies).