The set A in this case must be the convex hull of B. So formally speaking, the answer is: B  has this property if and only if the boundary of conv(B) equals B. Whether this answer is satisfactory for you or not, depends on the mathematical language in which you would like to express the answer. For example, the answer could be: B has this property if and only if it fits in one of two cases:  either B is closed convex and has empty interior, or B is an (n-1)-dimensional surface that separates Rn into two parts, and one of these parts is convex.

Dear Prof. Vladimir Kadets

Thanks a lot for your answer.

From geometric points of view, your answer is very good.

You are right, I did not mention what kinds of characterization may be satisfactory.

My purpose is mainly to find a good  parametrization function for boundary of convex sets. In a other word, finding a one-one, continuous ( and maybe convex in some sense ) map f: B --------> R^k    (  where k < n ). then we can use f(B) instead of B.

or finding a well shaped (  for example closed and arc-wise connected ) set like C in R^k such that  C is homeomorphic to B.

Thank you.

You can try the support function. Every convex set S  in X is characterized by its support function \sigma: X* \to R given by sigma(p) = sup{(p,x): x \in S} where X* is the dual space of X. If the maximum is attained in z it is not difficult to show that z belongs to the boundary of S. Also by Hahn-Banach theorem we have that clconv S = {x \in X: (p,x)\leq sigma(p) for all p \in X*}.

Perhaps, this will suit more for your purpose.

Thanks a lot Yura Malitsky.

Your answer is somewhat suitable.

According  to your answer it is easy to check that  bd(S) = {x \in X: (p,x) \leq sigma(p) for all p \in X*}  \   {x \in X: (p,x)  < sigma(p) for all p \in X*}.  where S is convex set in finite dimensional space.  the same assertion is sill true in infinite dimensional  spaces (however, in this case, I think  we should add weak  compactness assumption on S ). So  we obtained a characterization for boundary of convex sets  ( at least within finite dimensional spaces). That is good, however the support function does not give us  a parametrization  for these sets. I am seeking  to find well shaped ( for example closed and arc-wise connected ) set like C in R^k such that C is homeomorphic to bd(S).

Thank  you.

Dear Ashkan,

If your set is bounded, then the answer is easy, because the boundary of a bounded convex set A with non-empty interior is homeomorphic to the unit sphere of the space. Namely, if you shift A in such a way that 0 becomes its interior point, then the function x \mapsto x / ||x|| maps the boundary of A bijectively to the unit sphere, so the inverse function is the parametrization you are looking for.

All the best, Vladimir

Dear Prof. Vladimir Kadets

I truly appreciate  for this hint. That is what I was looking for.

I don't know why I didn't think about that function. 

regarding you answer,  in finite dimensional spaces,  this parametrization is a sill true for convex set with empty interior to characterize relative boundary  (because every convex set has non-empty relative interior) , and moreover in general cases ( even unbounded set) I think  every boundary of convex set is homeomorphic to  bd( B + C), where B is closed unite ball and C convex and closed cone.

Thank you.

The gauge function of the boundary of a convex set is a convex function. 

Not quite an answer to your question, but there is a elegant result by Richard B. Holmes http://www.ams.org/journals/tran/1973-184-00/S0002-9947-1973-0326252-2/S0002-9947-1973-0326252-2.pdf which roughly states that if the boundary of a closed convex set is regular (continuously differentiable a number of times) at a point, then the metric projection, the minkowski gauge function, and other natural convex functions one can define on the convex set are just as regular,  in a neighborhood of the normal outward ray at that point.

How are the homotopic properties of the boundaty of convex sets?

There are some papers by E. M. Bronshtein dealing with topological characterizations of the sets of extreme points of compact convex sets, both in finite and infinite dimensionsal.  Here are two of his papers, but he wrote several on this topic.

1. Bronshtejn, E.M.

On compact convex sets with given extreme points. (English. Russian original) Zbl 0883.46007

Sib. Math. J. 36, No.1, 17-23 (1995); translation from Sib. Mat. Zh. 36, No.1, 20-27 (1995).

2. Bronshtejn, E.M.

Topological properties of extreme points of convex compact sets in ℓ 2 . (English. Russian original) Zbl 0856.46004

Sb. Math. 186, No.3, 327-336 (1995); translation from Mat. Sb. 186, No.3, 19-28 (1995).

I suppose B should be n-1 dimensional, connected, without boundary, with all principal curvatures of the same sign. 

This is a good question.

A good place to start in answering this question is

S. Alexander, M. Ghomi, The convex hull property of non compact hyper surfaces with positive curvature:

http://www.math.uiuc.edu/~sba/nchp.pdf

Theorem 1.1,  page 1, is helpful.   Let M be a metrically complete, positively curved immersed hyper surface with compact boundary $\partial M$ and let

\begin{allgn*}

f: M &\longrightarrow \mathbb{R}^{n+1}\\

C  &= conv( f (\partial M ) ) \mbox{convex hull of image of boundary of M}\\

f(int M) \cap C &= \emptyset

\end{align*}

which says that the image of the interior of M lies completely outside the convex hull of the image of the boundary.    A neat result.     The proof of Theorem is given in Section 3, starting on page 3.   An example of an immersed surface is given in Fig. 1 (see the attached image).

More to the point, consider

U. Eckhardt, W. Scherl, Z. Yu, Representations of plane curves by means of descriptors in Hough space:

http://www.math.uni-hamburg.de/home/eckhardt/HOUGH.pdf

This thesis has a crisp way to denote the boundary of any set M in the plane ( bd M ).  See Chapter 3, starting on page 6, on the classification of descriptors in terms of a continuous, piecewise smooth curve K in the plane.   Then see Chapter 8, starting on page 17, on convex boundary segments.    In ch. 8, starting on page 20, a curve K is reconstructed up to boundary tangents.   This is an important chapter, since it considers the boundery of a convex set and boundary of the complement of a convex set.

A discussion on this question is included at the page 285 of the recent book by Valeriu Soltan, Lectures on Convex Sets, World Scientific 2015.

Let us denote Ashkan Mohammadi question as Question 1.

It seems that

Question 2.: How can one characterize the boundary of a strictly

convex set?

is related to the original question ( but it is easier then

Question 1).

For example, one can check the following:

If $A$ is a closed Jordan curve un plane, then the following

conditions are equivalent

(1) $A$ is the boundary of a strictly convex set

(2): every line intersect $A$ at most two points (two points

property).

(3): $A$ is union of two sub-arcs which are graphs of strictly

convex functions

We can generalize this result to several dimensions.

As soon as possible I will return to  the original question ( it seems useful to study the literature suggested by  James F Peters  and Stefan Cobzas).

Dear followers,

Suppose that

(i) $B$ is homeomorphic to $n-1$ dimensional sphere

and

(ii) for every point $x$ in $B$ there is a closed

half-spaces $L(x)$ (sets of point in space that lie on and to one

side of a hyperplane) which contains $B$.

It seems at a first glance that properties (i) and (ii)

guarantee hat there exist the closed convex set A such that

$\partial A=B$.

Please tell me if I miss something here.

u mean convex by algebraic or geomedric one