Tagged Questions

1
vote
1answer
49 views

Convex analysis: relative interior in finite and infinite dimension

Let $X$ be a normed space. Given a nonempty convex set $C \subset X$, the relative interior of $C$, denoted by $\text{ri} C$, is the set of the interior points of $C$, considered as a subset of ...
2
votes
2answers
58 views

Lipschitz functions in $\mathbb{R}^n, \ \ \mathbb{R}^m$, extension

I've found the following lemma : Let $\{x_1, . . . , x_k\}$ be a finite collection of points in $\mathbb{R}^n$ , and let $\{y_1, . . . , y_k\}$ be a collection of points in $\mathbb{R}^m$, such that ...
0
votes
1answer
75 views

Is this sum of convex and concave functions a convex function?

Is this a convex function in $X$, where all the entries are real and $Y,\beta$ are constants where $X,Y$ are rectangular matrices and $\beta$ is a constant vector and $A,B$ are constant p.s.d ...
0
votes
1answer
26 views

Continous map assuming positive value in the closure of a convex set

Let $\phi:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous map w.r.t. Euclidean norm. Let $C\subseteq\mathbb{R}^d$ be a convex subset. Assume that there exists $y\in\overline{C}$ such that ...
5
votes
0answers
115 views

Proof that the set of doubly-stochastic matrices forms a convex polytope?

Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to ...
2
votes
2answers
104 views

$ \text{int}(A) = \text{int(cl}(A))$, where $A$ is convex

How can one prove that $ \text{int}(A) = \text{int(cl}(A))$, where $ A \subseteq \mathbb{R}^n$ is convex?
4
votes
1answer
73 views

$ \text{cl}(\text{int}(A)) = \text{cl}(A)$ when A is convex

How can one prove that $ \text{cl}(\text{int}(A)) = \text{cl}(A)$, where $ A \subseteq \mathbb{R}$ is convex? I know that if $A$ is convex, $\text{int(A)}$ and $\text{cl(A)}$ are convex too. I need ...
2
votes
1answer
79 views

relative interior, convex hull, intersection

For any index set $I$, let $A_\iota\subseteq\mathbb{R}^d$ for $\iota\in I$ be closed sets. Do we have $\bigcap_{\iota\in ...
3
votes
1answer
96 views

Interior point and Minkowski functional

I want to prove for a convex set that contains zero a point $x$ is interior iff the value of Minkowski functional in that point is strictly less than $1$. is there anyone to help me.
3
votes
0answers
68 views

When is a sequentially closed cone, closed?

Let $X$ be a locally convex, Hausdorff topological vector space and $C\subseteq X$ a convex cone, which is sequentially closed. What are criteria, that would imply that $C$ is closed (in the topology ...
2
votes
2answers
93 views

Sets whose intersection with line segments have finite components

Certain subsets $A$ of $\mathbb{R}^n$ satisfy the property that given any line segment $L$, $A \cap L$ has a finite number of connected components. For instance, $A$ can be any finitary union of ...
2
votes
1answer
57 views

Convex functions and uniform convergence of derivatives

Let $f_n:[0,1]\to\mathbb{R},\ n\in\mathbb{N}$ be a sequence of convex analytic functions. Consider the sequence of derivatives $(f_n')_{n\in\mathbb{N}}$. Suppose that ...
0
votes
1answer
73 views

affine set convex set

How to show the following: Let $C$ be a convex subset of $\mathbb R^d$. Then $\operatorname{int} C \neq \emptyset$ if and only if $\operatorname{aff} C = \mathbb R^d$ where $\operatorname{aff} C$ is ...
0
votes
0answers
36 views

About the problem 20 chap 3 (functional analysis, Walter Rudin) [duplicate]

Possible Duplicate: I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed. Let $\{u_1,u_2,u_3,\dots \}$ be sequence of pairwise orthogonal unit vectors in Hilbert ...
1
vote
1answer
144 views

Prove that the spaces have the same homotopy type

This exercise was taken from the book "Fundamental Groups and Covering Spaces", from Elon Lages Lima. "Let $X=C_1\cup\cdots\cup C_k$ be a finite union of convex open sets in the Euclidean space ...
6
votes
1answer
148 views

Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems

There are some set systems with algebraic titles, such as "field", "algebra", "ring" and "semi-ring" (and possibly other titles), in their names. Examples are a sigma field (aka sigma algebra, ...
4
votes
1answer
159 views

When does it make sense to define a base of a set system?

In a topology, a base is defined to be a class of subsets such that every open set is the union of some members of it. In a convexity structure, a base is defined to be a class of subsets ...
2
votes
2answers
86 views

When does it make sense to define a generator of a set system?

In a set system, such as a topology, sigma algebra or convexity structure, a generator is defined to be a class of subsets such that the given set system is the coarsest such set system ...
1
vote
1answer
98 views

Is $S^\circ$ convex if $S$ convex?

Suppose $S\subset \mathbb{R^n}$ and $S^\circ$ denoted as the interior of $S$.Is $S^\circ$ convex if $S$ convex? $S$ is Convex mean $ \forall x,y\in S, kx+(1-k)y\in S, k\in [0,1]$ I know how to prove ...
2
votes
1answer
146 views

cantor intersection theorem in banach space

Here is part of the question in my HW. Let$\ \{C_n\}\subset X $ be a bounded nested decreasing sequence of closed and convex sets. I am asked to show that $\bigcap C_n \not= \emptyset$ iff X is ...
6
votes
1answer
302 views

Convex hull of an open set…

Let $K$ be a compact convex subset of locally convex topological vector space $E$. Let $U$ be an open subset of $K$. Is $conv(U)$ (the convex hull of $U$) an open subset of $K$ ? You see, it is ...
2
votes
3answers
111 views

Balanced but not convex?

In a topological vector space $X$, a subset $S$ is convex if \begin{equation}tS+(1-t)S\subset S\end{equation} for all $t\in (0,1)$. $S$ is balanced if \begin{equation}\alpha S\subset S\end{equation} ...
4
votes
3answers
159 views

Prove one set is a convex hull of another set

Define two sets: $A = \{x \in \{0,1\}^n : \lVert x \rVert_1 \leq k\}$ is a finite set of binary vectors; $B = \{x \in [0,1]^n : \lVert x \rVert_1 \leq k\}$ is an infinite set of real-valued vectors, ...
17
votes
1answer
549 views

Does local convexity imply global convexity?

Question: Under what circumstances does local convexity imply global convexity? Motivation: Classically, a twice differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ is convex if and only ...
17
votes
1answer
377 views

Is $[0,1]^\omega$ homeomorphic to $D^\omega$?

Let $n\in \mathbb N$ and let $D^n$ be the closed $1$-ball in $\left(\mathbb R^n, \|\,\cdot\,\|_1\right)$. It is not too hard to show that $[0,1]^n \cong D^n$ in this case. This observation leads to ...
1
vote
3answers
93 views

Reference for showing two closed and convex sets are equal

Suppose I have two sets $P, Q \subseteq R^d$ such that $P\subseteq Q$. $P$ and $Q$ are both convex and closed. I wish to show that $P=Q$. A straightfoward way to show this is showing $\forall y \in ...
2
votes
1answer
110 views

Generalizations of the Convex Hull

I am aware of generalizations of the convex kernel, via the addition of more polygonal line segments between points in a set. However, I wonder if there are similar generalizations for the convex ...
2
votes
1answer
272 views

Prove that the convex hull of a set is the smallest convex set containing that set

How do you prove that the convex hull of A is the smallest convex set containing A? edit: definition of a convex hull: Given a set A ⊆ ℝn the set of all convex combinations of points from A is called ...
11
votes
4answers
1k views

Why does a convex set have the same interior points as its closure?

Let $C$ be a convex subset of $\mathbb{R}^n$. I've been trying for hours to prove that $\dot{\overline{C}}=\dot{C}$. Somehow my intuition completely fails me. I found a proof in a textbook, but just ...