On smoothness, tangent cones, and the metric geometry of definable sets

TL;DR

This paper characterizes the $C^1$ smoothness of definable sets in o-minimal structures through tangent cones and metric properties, unifying several existing characterizations and providing new equivalences.

Contribution

It provides new definitive characterizations of $C^1$ smoothness for definable sets using tangent cones and metric conditions, extending previous results.

Findings

Equivalence of Lipschitz normal embedding and tangent cone conditions for definable sets.

Characterization of $C^1$ smoothness via tangent cone properties.

Unified framework connecting topological, metric, and smoothness properties.

Abstract

In this paper, we present several definitive characterizations of the $C^1$ smoothness of definable sets in terms of their tangent cones and some other metric properties. In particular, we recover some of the beautiful characterizations presented by Ghomi and Howard (2014) and by Kurdyka, Le Gal, and Nhan (2018). For instance, we prove that for any $X\subset \mathbb{R}^n$ that is a locally closed $d$-dimensional definable set in an o-minimal structure, the following items are equivalent: (1) $X$ is Lipschitz normally embedded (LNE), $C_3(X,p)$ is a $d$-dimensional linear subspace for any $p\in X$ and depends continuously on $p$; (2) For each $p\in X$, $X$ is Lipschitz regular at $p$ and $C_4(X,p)$ is a $d$-dimensional linear subspace; (3) $X$ is a topological manifold and for each $p\in X$, $X$ is LNE at $p$ and $C_4(X,p)=C_3(X,p)$; (4) $X$ is a topological manifold, and $C_5(X,p)$ is a $d-$dimensional subset for any $p\in X$; (5) $X$ is $C^1$ smooth.