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Let $(X,d)$ be a metric space, $x\in X$, and $Y\subset X$ is a closed set. Assume that $X$ is also a real vector space, so that we can form linear combinations over $\mathbb{R}$, but I do not assume ...

Note: This is a follow-up question inspired by a previous (more difficult) question I asked on MathOverflow. Let $f:\mathbb{R}\to\mathbb{R}$ be a (sufficiently regular, e.g. smooth) probability ...