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For example, if the convexity radius of a point x in a riemannian manifold M (without boundary) is R, what can we say about the convexity radius of points in B_R(x)?

The convexity radius of x is the sup of R s.t. B_R(x) is convex, and is always positive.

fritz
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  • Ok, I got some sort of an answer from klingenberg, riemannian geometry, page 85, corollary 1.9.11: the convexity radius (with a different and stronger definition) is lipschitz, with lipschitz constant at most 1! – fritz Apr 09 '14 at 17:01

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And the example on page 84 of that book shows that with your definition of convexity we cannot say much. Let $M$ be a very sharp cone, slightly rounded. Like this, only sharper.

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Let $x$ be the tip of the cone. Then $B_R(x)$ is convex for all $R$; the convexity radius is infinite. But for all other points $y$, apart from the tiny rounded end, the geodesic ball wraps around the cone before covering the tip. The closer is $y$ to $x$, the sooner this happens (until we enter the tiny rounded end). So the radius of convexity at $y$ can be extremely small.