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Let $\Sigma$ be an Alexandrov space of curvature $\geq 1$ without boundary. Let $X$ be the cone over $\Sigma$. $X$ is well known to be non-negatively curved.

Let $f\colon X\to \mathbb{R}$ be a concave function (i.e. its restriction to any shortest path is concave in the usual sense). Assume in addition that $f$ is 1-homogeneous and $f\geq 0$. Does it follow that $f\equiv 0$?

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