Intended for questions about convexity spaces: convex hulls; convexity preserving and convex-to-convex functions; isomorphism of convexity spaces; (Chepoi's) separation axioms S1, S2, S3, and S4; interval convexities; subbasis and basis of convexities and topics alike. When considering (usual) convex sets in vector spaces, please use the [convex-analysis] tag.
Questions tagged [convexity-spaces]
15 questions
11
votes
1 answer
Topology basis consisting of convex sets in metric spaces
Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
We then say that a set $S\subseteq X$ is convex if…
Tian Vlasic
- 1,436
8
votes
2 answers
Are convex polytopes closed in arbitrary metric spaces?
Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
We then say that a set $S\subseteq X$ is convex if…
Tian Vlasic
- 1,436
7
votes
2 answers
Are metric segments convex?
For a metric space $(X,d)$ and points $x,y \in X$ we define the metric segment between them as the following set:
$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$
Can we say that metric segments are convex? That is, for an…
Tian Vlasic
- 1,436
4
votes
1 answer
How do abstract convexity spaces generalise convex sets?
I am reading this paper where the definition of a convexity space is given as follows (page $3$ of paper).
Def 1: A convexity space on a set $V$ is a collection $C\subseteq2^{|V|}$ satisfying
$\emptyset, V\in C$,
$A, B\in C$ implies $A\cap B \in…
Tom Finet
- 603
3
votes
0 answers
Homeomorphism of $\mathbb{D}$ and convexity
I think this should be true.
Let $\varphi: \mathbb{D} \rightarrow \mathbb{D}$ be an homeomorphism such that $\varphi(0) = 0$. Then there exists $r < 1$ such that $\varphi(\mathbb{D}_r)$ is convex. $(\mathbb{D}_r = \{z \in \mathbb{C} \mid |z| \leq…
Alejandro
- 65
3
votes
1 answer
Convex hull of open sets is an open set?
Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
We then say that a set $S\subseteq X$ is convex if…
Tian Vlasic
- 1,436
3
votes
1 answer
Closure and interior of convex set is convex?
For a metric space $(X,d)$ and points $x,y \in X$ we define the metric segment between them as the following set:
$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$
We then say that a set $S\subseteq X$ is convex if for all…
Tian Vlasic
- 1,436
2
votes
0 answers
Generalization of Geometric Convexity Which Allows Saddle-Point-Like Structures
I've been thinking lately about a certain generalization of convexity. I'm sure this concept has been studied, but I don't know what it's called. Could you help me find its name?
Convexity in geometry is usually defined by saying that, for any two…
Martin Wickham
- 21
- 2
2
votes
1 answer
Convexity structures and partial orders
Can any convexity structure be defined by a partial order $\preceq$ in the sense of the order topology: a given set $A$ is convex if for any $a,b \in A$ and any other element $c$ for which $a\preceq c \preceq b$, we have that $c\in A$?
I feel like…
user146125
- 667
1
vote
0 answers
Examples of convexity spaces of arity $n$ that arise naturally (for large $n$, or infinite $n$)
The arity of a (non-empty) convexity space $ \left(X, \mathscr{C} \right) $
is defined as follow: $$ 0 \lt ar\left(\mathscr{C}\right) \le n \iff \left( C \in \mathscr{C} \iff\ C = \bigcup \{co\left(F\right) \vert F \subseteq C;\ card(F) \le n \}…
Pielcq
- 330
0
votes
0 answers
Fix point of an affine map
Let $E$ be a normed vector Space
Let $T: E \to E $ an affine map.
Let $C$ be a convex compact non empty of $E$ such that $T(C) \subseteq C$.
Show that $T$ fixes a point of $C$
I am stucked on this problem because $T$ is not necessarily continuous so…
Sine
- 1
0
votes
0 answers
Is there a commonly used term for these properties of extreme subsets?
Suppose we are working a real vector space $X$ with convex hull operator $\mathsf{con}$. Let $C$ be a convex set and $E$ an extreme subset of $C$. It is easy to check that if $e \in E$ and $c \in C \smallsetminus E$, then for all $x \in…
Jay
- 3,942
0
votes
1 answer
Reflexive and strictly convex but not uniform convex
I’m struggling with one question, I can find that authors are writing over uniform convex in Banach spaces a lot but still I haven’t found a good exampel for this:
If space X is reflexive and strictly convex then this is not implying uniform…
Dawid
- 1
0
votes
0 answers
Is a closed set in a TVS over $\mathbb{R}$ convex?
From Theory of Convex Structures by M. L. J. Van De Vel, on a set $X$, a topology and a convexity structure are said to be compatible, if the convexity structure is generated by the closed sets. The condition is equivalent to that all the polytopes…
Tim
- 49,162
0
votes
2 answers
How do I show that the open rectangle is convex?
I have no idea how to solve this. It's easy to show that the open ball or the open cube is convex, but how do we show that the open rectangle is? (The same holds for the closed rectangle).
Thanks
L.F. Cavenaghi
- 4,781