= Nearest point in a uniformly convex Banach space
Every nonempty closed convex subset of a uniformly convex <Banach space> has a unique nearest point to each vector. A minimizing sequence is Cauchy: otherwise uniform convexity would put some midpoint strictly closer than the infimum. Completeness and closedness give its limiting point. In $\ell^p$, $p\geq2$, the <Clarkson inequality> makes this quantitative: the distance between two minimizing vectors tends to zero after subtracting the midpoint's lower distance bound. Nonemptiness is required.
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