Title: Local Estimate on Convexity Radius and decay of injectivity radius in a Riemannian manifold,
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.