Affine invariant points and new constructions

TL;DR

This paper provides an alternative proof to a problem about affine invariant points of convex bodies and introduces a new large class of such points, advancing the understanding of affine geometry.

Contribution

It offers a new proof of a known result and constructs a broad new class of affine invariant points, enhancing the theoretical framework.

Findings

Confirmed the set of affine invariant points equals the set of points invariant under all affine automorphisms.

Developed an alternative proof method based on previous ideas.

Constructed a new large class of affine invariant points.

Abstract

In \cite{branko} Gr{\"u}nbaum asked if the set of all affine invariant points of a given convex body is equal to the set of all points invariant under every affine automorphism of the body. In \cite{ivan} we have proven the case of a body with no nontrivial affine automorphisms. After some partial results (\cite{aip},\cite{newaip}) the problem was solved in positive by Mordhorst \cite{Olaf}. In this note we provide an alternative proof of the affirmative answer, developing the ideas of \cite{ivan}. Moreover, our approach allows us to construct a new large class of affine invariant points.