Lipschitz geometry of pairs of normally embedded H\"older triangles

TL;DR

This paper introduces a combinatorial invariant called στ-pizza to classify pairs of normally embedded H"older triangles under outer bi-Lipschitz equivalence, advancing understanding of semialgebraic surface germs.

Contribution

It defines the στ-pizza invariant and conjectures its completeness for classifying certain surface germs under bi-Lipschitz transformations.

Findings

Proposes the στ-pizza as a new invariant.

Conjectures the invariant's completeness for classification.

Focuses on pairs of H"older triangles in surface germs.

Abstract

We consider a special case of the outer bi-Lipschitz classification of real semialgebraic (or, more general, definable in a polynomially bounded o-minimal structure) surface germs, obtained as a union of two normally embedded H\"older triangles. We define a combinatorial invariant of an equivalence class of such surface germs, called $\sigma\tau$-pizza, and conjecture that, in this special case, it is a complete combinatorial invariant of outer bi-Lipschitz equivalence.