0
votes
0answers
82 views

ellipsoids have spherical section

I want to prove that "For any $(2k-1)$-dimensional ellipsoid $E$ ,there is a $k$-flat $L$ passing through the center of $E$ such that $ E \cap L$ is a Euclidean ball. I see a proof for it in the book ...
2
votes
1answer
180 views

Chebyshev centres of a bounded closed convex set in a strictly convex Banach space

Suppose $X$ is a strictly convex Banach space. Does there exist a bounded closed convex set $K$ in $X$ such that the set of all Chebyshev centers $C(K)$ of $K$ is a proper subset of $K$ with diameter ...
3
votes
1answer
220 views

Characterization of $l_p$ up to a linear isometry

There is a science called "Geometry of Banach spaces". I wonder if they managed to give a geometric characterization of $\ell_p$ ($p\in[1,\infty]$) up to isometric isomorphism (among all Banach ...
13
votes
2answers
541 views

Intuitive explanation of Dvoretzky's theorem

I am wondering if anyone has an enlightening explanation of why Dvoretzky's theorem (which says that a high-dimensional convex body has an almost round central section) is true -- there are a number ...
1
vote
0answers
89 views

Extenstions of Urysohn's inequality

A version of Urysohn's inequality states that for a symmetric convex body $K \subset \mathbb{R}^n$, one has $$ \left(\frac{\text{vol}(K)}{\text{vol}(B_2)} \right)^{1/n} \le \frac{1}{\sqrt{n}} E \; \| ...
7
votes
2answers
268 views

Duality between extremal points and extremal maps

Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set ...
2
votes
2answers
385 views

Finite imensional subspaces of $L^\infty.$

This is the question I had meant to ask when I asked this question.: Is there a concise characterization of finite dimensional subspaces of $L^\infty?$ (that's what the discussion with @fedja was ...
5
votes
3answers
296 views

Finite dimensional subspaces of $L^1.$

This question is motivated by my discussion (via comments) with @fedja regarding this earlier question. In any case the question is whether there is any concise characterization of finite dimensional ...
2
votes
0answers
144 views

Parameterizing infinitesimal perturbations of the sphere using signed measures

Let $\mathcal K^3$ be the space of convex bodies, with some metric $\delta$, and let $B$ be the unit ball. Let us define a volume-preserving perturbation of the sphere to be a continuous (substitute ...
5
votes
0answers
260 views

Local minimum from directional derivatives in the space of convex bodies

I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...
4
votes
0answers
138 views

Relationship between sequential compactness of a convex set and its extremal points

Suppose that $X$ is a compact convex subset of a topological vector space. Suppose also that the extremal points of $X$ have the additional property that any sequence $x_n$ of extremal points has a ...
8
votes
1answer
557 views

volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

We denote by $\otimes_{\epsilon}$ the injective Banach tensor product. Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
1
vote
1answer
295 views

How do maximum norms relatively change in Euclidean translations

Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$ that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible linear transformation from ...
1
vote
0answers
386 views

Does the dual Banach space $B(\ell^\infty)$ have weak* normal structure?

Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if $$ \sup_{y\in K} \|x-y\|={\rm diam}(K). $$ where ${\rm diam}(K)$ denotes the ...
11
votes
3answers
624 views

Continuous automorphism groups of normed vector spaces?

Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...