1
vote
0answers
111 views

Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$): $\min_{x\in U}\|x-y\|$. I believe the minimizer $x_{U}^{*}$ is ...
2
votes
1answer
80 views

Convex interaction energy

Does anybody know examples of absolutely continuous probability measures $\mu_0,\mu_1$ on $\mathbb{R}^n$ with finite 2nd moments such that $$ \frac{d^2}{dt^2}\left(\int_{\mathbb{R}^n\times ...
2
votes
5answers
208 views

Distance between two sets

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem. $$ \min ...
2
votes
2answers
279 views

convergence of the infima of convex functions

Can one give a reference to a result like this: If a sequence of convex functions $f_{n}$ on $\mathbb{R}$ converges pointwise to a non-monotonic function $f$, then ...