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The minimum of $J$ is not unique $m=3$, $n=4$, $E=\{(1,2),(1,3),(2,4),(3,4)\}$ and $X=\mathbb R^2$. Assume that $a_1$, $a_2$ and $a_3$ are vertices of huge equilateral triangle. Then at the minimum o …

The answer is "no". Suppose $D$ be a disc with one handle, nonpositive curvature, geodesic boundary with flat collar. Note that there is a geodesic $\gamma\colon[0,\infty)\to D$ that starts at $\parti …

The answer is "no". Let $\Sigma$ be the suspension over Poincaré homology sphere. It admits a polyhedral $\mathrm{CAT}[1]$-metric. Let $B$ be the unit ball in the Euclidean cone $\mathrm{Cone}\,\Sig …

It meant to be an answer to another question, https://mathoverflow.net/a/406833/, but it is an answer to this question as well.

Let me describe a 6-point counterexample. Let $K$ be a 2-dimensional cone with total angle $\theta=2{\cdot}\pi+\varepsilon$, where $\varepsilon$ is small and positive (any $0<\varepsilon<\tfrac\pi2$ w …

The answer is "no" even for 3-dimensional Hadamard manifolds. Moreover, implication $$\measuredangle[a\,^b_c]<\tfrac\pi2 \ \&\ \measuredangle[a\,^b_d]<\tfrac\pi2\ \Rightarrow\ \measuredangle[a\,^b_x] …