# Tagged Questions

**3**

votes

**1**answer

164 views

### Separable Banach spaces which are absolute Lipschitz retracts

A subset $F$ of a metric space $M$ is called a Lipschitz retract of $M$ if there is a Lipschitz map from $M$ onto $F$ which coincides with the identity on $F$. A metric space which is a Lipschitz ...

**13**

votes

**0**answers

380 views

### An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...

**4**

votes

**0**answers

107 views

### Hilbert metric of a sum of cones

Suppose that $K_{1}$ and $K_{2}$ are pointed closed cones in a finite-dimensional space $V$ whose Hilbert metrics $d_{1},d_{2}$ are known. Is there a way to express the Hilbert metric of $K_{1}+K_{2}$ ...

**3**

votes

**1**answer

99 views

### What is the doubling dimension of convex functions?

I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the ...

**0**

votes

**0**answers

116 views

### Gaussian width (or metric entropy) for the intersection of the $\ell_1$ and $\ell_2$ balls

Let $B_p := \{ x \in \mathbb{R}^d:\; \|x\|_p \le 1\}$ where
$\|x\|_p := (\sum_{i=1}^d |x_i|^p)^{1/p}$ is the $\ell_p$ norm.
(1) Let $t \in (0,1)$. Can we give an estimate on $$\mathbb{E} ...

**15**

votes

**0**answers

525 views

### Lipschitz Homeomorphisms Between Spheres of N-dimensional Spaces

Let $B_p^N$ be the unit ball of $\mathbb{R}^N$ under the $\ell_p^N$ norm.
Question: Let $C_N$ be the infimum of all $C$ for which there is a homeomorphism $f_N$ from $B_\infty^N$ onto $B_2^N$ so ...

**7**

votes

**1**answer

327 views

### Nonexpansive multi-valued maps in $\ell^2$

Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le ...

**4**

votes

**1**answer

192 views

### Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets

Given two $n-$dimensional convex compact sets $A,B$, we define $d(A,B)$ as $\log({\mathrm{Vol}}(\alpha_2(A)))-\log(\mathrm{Vol}(\alpha_1(A)))$ where $\alpha_1,\alpha_2$ are two affine bijections such ...