Arc criterion of normal embedding

TL;DR

This paper introduces a criterion for local normal embedding of semialgebraic sets based on comparing inner and outer contact orders of arcs approaching a point.

Contribution

It establishes a new arc-based criterion for normal embedding in semialgebraic and o-minimal structures, linking contact orders to embedding properties.

Findings

Normal embedding characterized by arc contact orders

Inner and outer contact orders are equal for normally embedded sets

Criterion applies to semialgebraic and definable sets in o-minimal structures

Abstract

We present a criterion of local Normal Embedding of a semialgebraic (or definable in an o-minimal structure) contained in $R^n$ in terms orders of contact of arcs. Namely, we prove that a semialgebraic set is normally embedded at a point x if and only if for any pair of arcs, coming to this point the inner order of contact is equal to the outer order of contact.