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

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.