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We show that for any $1\leq p\leq\infty$, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of $\ell_p^n$ verify the variance conjecture $$ \textrm{Var}\,|X|^2\leq C\max_{\xi\in S^{n-1}}\mathbb{E}\langle X,\xi\rangle^2\mathbb{E}|X|^2, $$ where $C$ depends on $p$ but not on the dimension $n$ or the hyperplane. We will also show a general result relating the variance conjecture for a random vector uniformly distributed on an isotropic convex body and the variance conjecture for a random vector uniformly distributed on any Steiner symmetrization of it. As a consequence we will have that the class of random vectors uniformly distributed on any Steiner symmetrization of an $\ell_p^n$-ball verify the variance conjecture.

We show that any random vector uniformly distributed on any hyperplane projection of $B_1^n$ or $B_\infty^n$ verifies the variance conjecture $$\text{Var}|X|^2\leq C\sup_{\xi\in S^{n-1}}\E^2\E|X|^2.$$ Furthermore, a random vector uniformly distributed on a hyperplane projection of $B_\infty^n$ verifies a negative square correlation property and consequently any of its linear images verifies the variance conjecture.

We recall two approaches to recent improvements of the classical Sobolev inequality. The first one follows the point of view of Real Analysis, while the second one relies on tools from Convex Geometry. In this paper we prove a (sharp) connection between them.

The isotropy constant of any $d$-dimensional polytope with $n$ vertices is bounded by $C \sqrt{n/d}$ where $C>0$ is a numerical constant.

We consider $k$-dimensional central sections of the unit ball of $\ell_p^n$ (denoted $B_p^n$) and we prove that their volume are bounded by the volume of $B_p^n$ whenever $1

The classical Brunn-Minkowski inequality states that for $A_1,A_2\subset\R^n$ compact, $$ |A_1+A_2|^{1/n}\ge |A_1|^{1/n}+|A_2|^{1/n}\eqno(1) $$ where $|\cdot|$ denotes the Lebesgue measure on $\R^n$. In 1986 V. Milman {\bf [Mil 1]} discovered that if $B_1$ and $B_2$ are balls there is always a relative position of $B_1$ and $B_2$ for which a perturbed inverse of $(1)$ holds. More precisely:\lq\lq{\sl There exists a constant $C>0$ such that for all $n\in\N$ and any balls $B_1,B_2\subset\R^n$ we can find a linear transformation $u\colon\R^n\to\R^n$ with $|{\rm det}(u)|=1$ and $$|u(B_1)+B_2|^{1/n}\le C(|B_1|^{1/n}+|B_2|^{1/n})"$$} The aim of this paper is to extend this Milman's result to a larger class of sets.

In this note we investigate some aspects of the local structure of finite dimensional $p$-Banach spaces. The well known theorem of Gluskin gives a sharp lower bound of the diameter of the Minkowski compactum. In [Gl] it is proved that diam$({\cal M}_n^1)\geq cn$ for some absolute constant $c$. Our purpose is to study this problem in the $p$-convex setting. In [Pe], T. Peck gave an upper bound of the diameter of ${\cal M}_n^p$, the class of all $n$-dimensional $p$-normed spaces, namely, diam$({\cal M}_n^p)\leq n^{2/p-1}$. We will show that such bound is optimum.

In this paper, equivalence between interpolation properties of linear operators and monotonicity conditions are studied, for a pair $(X_0,X_1)$ of rearrangement invariant quasi Banach spaces, when the extreme spaces of the interpolation are $L^\infty$ and a pair $(A_0,A_1)$ under some assumptions. Weak and restricted weak intermediate spaces fall in our context. Applications to classical Lorentz and Lorentz-Orlicz spaces are given.