Toward canonical convex functions in Alexandrov spaces

TL;DR

This paper constructs canonical convex functions in finite-dimensional Alexandrov spaces that approximate squared distance functions and can be extended to nearby spaces, aiding geometric analysis.

Contribution

It introduces a method to build 2-convex functions approximating squared distances in Alexandrov spaces, extendable to Gromov-Hausdorff close spaces.

Findings

Constructed 2-convex functions approximating squared distances.

Functions can be lifted to Gromov-Hausdorff close Alexandrov spaces.

Provides tools for geometric analysis in Alexandrov spaces.

Abstract

We construct for every finite-dimensional Alexandrov space $A$ and every point $p \in A$ a $2$-convex function $f_p$ in a small neighborhood around $p$, which approximates $\operatorname{dist}_p^2$ up to second order. Moreover, the function $f_p$ can be lifted to Gromov-Hausdorff close Alexandrov spaces of the same dimension.