Lipschitz geometry of pairs of normally embedded Hölder triangles

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 Hölder triangles. We define a combinatorial invariant of an equivalence class of such surface germs, called $$\sigma \tau $$ σ τ -pizza, and conjecture that, in this special case, it is a complete combinatorial invariant of outer bi-Lipschitz equivalence.

On Lipschitz Geometry at infinity of complex analytic sets

“…An example of such a relation was presented by Fernandes and the author of this article in [12]. By using the ideas of [1], the authors of [12] presented a global classification of semi-algebraic surfaces with isolated singularities under bi-Lipschitz homeomorphisms, with respect to its inner distance (socalled inner lipeomorphims). As a consequence, they obtained the following result (see definitions of stereographic modification and stereographic compactification of a set in Subsection 2.1): Theorem 1.1 (Corollary 5.7 in [12]).…”