# calculus – Topological properties for convex sets in R^n and projection under Bregman divergence

Assume $$D subset R^n$$ is an open and convex set, $$X subset R^n$$ is a convex set, assume $$D cap X ne emptyset$$, $$X subset bar{D}$$. For example, $$D = R^n_{++}$$ and $$X$$ is the $$n$$-simplex $$Delta_n = {x in R^n| sum_i x_i = 1, x_i ge 0}$$, I want to prove the following two facts.

Firstly, for any $$x_0 in X$$, there is a sequence of points $$x_1,x_2,dots$$ in $$Dcap X$$ converges to $$x_0$$ (in Euclidean norm), i.e. $$x_n rightarrow x_0$$.

The second fact is a little more complicated. Take a function $$Phi$$ on $$D$$, assume $$Phi$$ satisfying 2 properties: 1, $$Phi$$ is strictly convex and continuous differentiable. 2, $$| nabla Phi(x) | rightarrow infty$$ while $$x rightarrow partial D$$. Now take an arbitrary point $$y in D$$, prove the projection of $$y$$ on $$X cap D$$ under Bregman divergence exists. That is, the function $$D_{Phi}(x,y) = Phi(x) – Phi(y) – nabla Phi(y) cdot (x-y)$$ with domain $$x in D cap X$$ attains its minimum.