# Tangent cones and $C^1$ regularity of definable sets

## TL;DR

This paper characterizes when a connected definable set in an o-minimal structure is a $C^1$ manifold by examining tangent cones, their continuity, and density conditions, establishing equivalences among these geometric properties.

## Contribution

It proves the equivalence of three geometric conditions that characterize $C^1$ regularity for definable sets in o-minimal structures.

## Key findings

- Tangent cone and paratangent cone coincide at all points for $C^1$ manifolds.
- Tangent cone varies continuously and has constant dimension on $X$.
- Density $ heta(X, x) < 3/2$ characterizes smoothness.

## Abstract

Let $X\subset \mathbb R^n$ be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \in X$, the tangent cone of $X$ at the point $x$ is a $k$-dimensional linear subspace of $\mathbb R^n$ ($k$ does not depend on $x$) varies continuously in $x$, and the density $\theta(X, x) < 3/2$.