How to characterize the boundary of a convex set? I am working on a part of a paper related to topological properties of boundary points. It is important to me to realize the topological and algebraic behavior the boundary points of convex sets. I would be grateful if someone help me around these issues by giving some ideas or references related to it.
The general question provided in following;  
Let $B$ be a closed set in $n-$dimensional Euclidean space. What other properties $B$ should have in order to be guaranteed that there is closed convex set $A$ such that $\partial A = B. $ 
How about infinite dimensional spaces?    
 A: In general, chapter A.2 and A.4 of the book "Hiriart-Urruty, Jean-Baptiste, and Claude Lemaréchal. Fundamentals of convex analysis." might be useful reading, depending on what type of characterization you want.
My answer is for an n-dimensional Euclidean space, I am not sure how well everything generalizes to infinite-dimensional spaces.
Two facts that might be useful:


*

*For any convex set $C\in \mathbb{R}^n, C\ne \emptyset, C\ne \mathbb{R}^n$, a point $x\in C$ is on the boundary if and only if there is a  hyperplane supporting $C$ at $x$ (page 44 as well as Lemma 4.2.1 in the reference above).

*A closed convex set can be defined as the intersection of all closed half-spaces that contain it (Theorem 4.2.3 and Corrolary 4.2.4 in the reference).
One conclusion that we can draw from this is that $B$ has the desired property if and only if $B$ is the boundary of an intersection of half-spaces (assuming that $B$ is nonempty, I didn't think about dealing with $C=\mathbb{R}^n$ and $C=\emptyset$ here).
