# Local Estimate on Convexity Radius and decay of injectivity radius in a Riemannian manifold

## TL;DR

This paper establishes curvature-free, pointwise estimates for convexity and injectivity radii in Riemannian manifolds, providing insights into local geodesic behavior and clarifying concepts of convexity radius.

## Contribution

It introduces new curvature-free estimates for convexity and injectivity radii and clarifies the concept of convexity radius in Riemannian geometry.

## Key findings

- Convexity radius bounds involving injectivity and focal radii.
- Injectivity radius comparison based on conjugate points.
- Geodesic length bounds within convex neighborhoods.

## Abstract

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname{inj}(p)}(p))\}$, where $\operatorname{inj}(p)$ is the injectivity radius of $p$ and $\operatorname{foc}(B_r(p))$ is the focal radius of open ball centered at $p$ with radius $r$; 2) for any two points $p,q$ in $M$, $\operatorname{inj}(q)\ge \min\{\operatorname{inj}(p), \operatorname{conj}(q)\}-d(p,q),$ where $\operatorname{conj}(q)$ is the conjugate radius of $q$; 3) for any $0<r<\min\{\operatorname{inj}(p),\frac{1}{2}\operatorname{conj}(B_{\operatorname{inj}(p)}(p))\}$, any (not necessarily minimizing) geodesic in $B_r(p)$ has length $\le 2r$. We also clarify two different concepts on convexity radius and give examples to illustrate that the one more frequently used in literature is not continuous.