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Let $X$ be a $\mathrm{CAT}(0)$ space, $p\in X$ and $v\in T_pX$. Let $N\subset T_pX$ be the set of tagent vectors making an angle greater than or equal to $\pi/2$ with $v$.

Is it true that the set $\exp(N)\subset X$ is convex (that is, for every couple of points, the minimal geodesic between them is contained in $\exp(N)$)?

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This is not true even at an ordinary conical singularity with total angle greater than $2\pi$ on a surface.

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