Affine functions on Alexandrov spaces

TL;DR

This paper demonstrates that finite-dimensional Alexandrov spaces with curvature bounds can be embedded into a product space, where affine functions on the original space correspond to affine functions on Euclidean factors, revealing structural insights.

Contribution

It introduces a canonical embedding of finite-dimensional Alexandrov spaces into product spaces, linking affine functions to Euclidean components, a novel structural result.

Findings

Finite-dimensional Alexandrov spaces embed into product spaces.

Affine functions on these spaces originate from Euclidean factors.

The embedding preserves affine function structures.

Abstract

We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.