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Let $ S $ be the $ m $ dimensional unit sphere in $ \mathbf{R}^{m+1} $ and let $ B $ be a closed ball in $ \mathbf{R}^{m+1} $ such that $ B \cap S $ lies within an open hemisphere of $ S $. Is $ B \cap S $ geodesically convex in $ S $?

It is not difficult to see that there are convex bodies $ B $ with smooth boundary such that $ B \cap S $ lies within an open hemisphere but $ B \cap S $ is not geodesically convex.

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    $\begingroup$ Yes. In particular the boundary $\partial B\cap S$ is a sphere of dimension $m-1$. You get this by the axial symmetry of the entire set-up with respect to the line joining the centers of $S$ and $B$. $\endgroup$
    – Willie Wong
    Commented Sep 7, 2016 at 20:49

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