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.