# Tagged Questions

**0**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**3**answers

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

**0**answers

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**

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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 ...

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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

**1**answer

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

**1**answer

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

**0**answers

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

**3**answers

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 ...