Lipschitz geometry and combinatorics of abnormal surface germs

We study outer Lipschitz geometry of real semialgebraic or, more general, definable in a polynomially bounded o-minimal structure over the reals, surface germs. In particular, any Hölder triangle is either normally embedded or contains some “abnormal” arcs. show that abnormal arcs constitute finitely many “abnormal zones” space all arcs, and investigate geometric combinatorial properties establish strong relation between combinatorics triangles.