I have 2 questions about the fundamental properties of mirror descent.

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 $D\cap X$ 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.


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