Questions tagged [convexity-spaces]
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.
12
questions
2
votes
1
answer
92
views
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 ...
- 438
3
votes
0
answers
61
views
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. $...
- 65
0
votes
1
answer
51
views
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 ...
- 1
0
votes
0
answers
40
views
Convexity of a scaled multivariate digamma function
The problem...
Let $\psi_p(a) = \frac{\partial \Gamma_p(a)}{\partial a}$ be the multivariate digamma function. I believe the following function to be strictly convex at least for real values $a>2p$...
8
votes
2
answers
208
views
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:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
...
- 1,289
2
votes
1
answer
323
views
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:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
...
- 1,289
10
votes
1
answer
383
views
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:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
...
- 1,289
2
votes
1
answer
145
views
Family of sets closed under arbitrary intersections and arbitrary unions of chains of its elements induces a finitary closure operator?
Let $\mathcal{U} \subseteq \wp\left ( X \right )$ be a family of sets that contains both $\emptyset$ and $X$, is closed under arbitrary intersections and is closed under arbitrary unions of chains of ...
- 1,289
3
votes
1
answer
546
views
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 ...
- 1,289
6
votes
2
answers
376
views
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 ...
- 1,289
3
votes
1
answer
87
views
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
$\...
- 593
0
votes
2
answers
2k
views
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
- 4,661