Let U, V be two symmetric convex bodies in ${ℝ}^n$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors $u_1,...,u_n\in U$ such that, for each choice of signs ${\epsilon }_1,...,{\epsilon }_n=\pm 1$, one has ${\epsilon }_1{u}_1+...+{\epsilon }_n{u}_n\notin rV$ where $r={\left(2\pi e^2\right)}^{-1/2}{n}^{1/2}\left(\right|U|/{\left|V\right|)}^{1/n}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $\left(u_n\right)$ such that the series ${\sum }_{n=1}^{\infty }{\epsilon }_n{u}_{\pi \left(n\right)}$ is divergent for any choice of signs ${\epsilon }_n=\pm 1$ and any permutation π of indices.

We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$-dimensional case, our main contribution is a method for determining the optimal coefficients $c_1,...,c_m$ in the inequality $\delta P\left(E\right)\ge {\sum }_{k=1}^m{c}_k\alpha {\left(E\right)}^k+o\left(\alpha {\left(E\right)}^m\right)$, valid for each Borel set $E$ with positive and finite area, with $\delta P\left(E\right)$ and $\alpha \left(E\right)$ being, respectively, the $\mathrm{𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡}$ and the $\mathrm{𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦}$ of $E$. In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of $\mathrm{𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠}$ including the lower semicontinuous extension of $\frac{\delta P\left(E\right)}{\alpha {\left(E\right)}^2}$, we describe the...

We prove an almost isometric reverse Hölder inequality for the euclidean norm on an isotropic generalized Orlicz ball which interpolates Paouris concentration inequality and variance conjecture. We study in this direction the case of isotropic convex bodies with an unconditional basis and the case of general convex bodies.

On définit une structure de bigèbre différentielle graduée sur la somme directe des complexes cellulaires des permutoèdres, qui contient une sous-bigèbre différentielle graduée dont le complexe sous-jacent est la somme directe des complexes cellulaires des polytopes de Stasheff. Ceci étend des constructions de Malvenuto et Reutenauer et de Loday et Ronco pour les sommets des mêmes polytopes.

A generalization of the theorem of Bajmóczy and Bárány which in turn is a common generalization of Borsuk's and Radon's theorem is presented. A related conjecture is formulated.