Timeline for Metric projection on closed convex sets in Busemann space

6 events

when toggle format	what		by	license	comment
Apr 14, 2021 at 15:11	answer	added	Logan Fox		timeline score: 2
Jan 9, 2021 at 20:59	history	edited	Logan Fox	CC BY-SA 4.0	added link to proposition in question
Jan 9, 2021 at 0:53	comment	added	JHM		The infimum $\inf_{a\in A} d(x,a)$ obviously exists since $d\geq 0$. If $d$ is proper, then every minimizing sequence will be contained in a compact ball, and there will exist a convergent subsequence. Without $d$ proper, there is another possibility, using fact that convex hulls of finite subsets is always compact in nonpositive curvature (convex hull of finite subset is the continuous image of a compact simplex). If $d$ nonproper and $A$ noncompact, then i think minimum possibly does not exist.
Jan 8, 2021 at 18:06	comment	added	Logan Fox		@JHM Using the strict convexity of $d^2$ makes sense for uniqueness. It's more the existence part that is evading me.
Jan 8, 2021 at 0:46	comment	added	JHM		If I'm not mistaken, one can argue that $d^2/2$ is strictly convex if $d$ is convex. And with strict convexity, for every $x$ we deduce the uniqueness of $argmin_{a\in A} d^2(x,a)/2$, from which uniqueness of $argmin_{a\in A} d(x,a)$ follows. The strict convexity of quadratic distance $d^2/2$ versus the convexity of $d$ is basically why $L^2$ optimal transport is so regular and $L^1$ optimal transport more difficult.
Jan 7, 2021 at 23:58	history	asked	Logan Fox	CC BY-SA 4.0