arXiv subjects
Publications and source records attributed to Grigoris Paouris.
The purpose of this article is to describe a reduction of the slicing problem to the study of the parameter I_1(K,Z_q^o(K))=\int_K ||< :, x> ||_{L_q(K)}dx. We show that an upper bound of the form I_1(K,Z_q^o(K))\leq C_1q^s\sqrt{n}L_K^2, with 1/2\leq s\leq 1, leads to the estimate L_n\leq \frac{C_2\sqrt[4]{n}log(n)} {q^{(1-s)/2}}, where L_n:= max {L_K : K is an isotropic convex body in R^n}.
We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in R^n by its dual L_{p$-centroid bodies is independent of the geometry of P. In particular we show that if P has volume 1, lim_{p\rightarrow \infty} \frac{p}{\log{p}} (\frac{|Z_{p}^{\circ}(P)|}{|P^{\circ}|} -1) = n^{2}. We provide an application to the approximation of polytopes by uniformly convex sets.
Let $K$ be a convex body in $\mathbb R^n$. We introduce a new affine invariant, which we call $Ω_K$, that can be found in three different ways: as a limit of normalized $L_p$-affine surface areas, as the relative entropy of the cone measure of $K$ and the cone measure of $K^\circ$, as the limit of the volume difference of $K$ and $L_p$-centroid bodies. We investigate properties of $Ω_K$ and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show a "information inequality" for convex bodies.
We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If $K$ is a convex body in ${\mathbb R}^n$ with volume one and center of mass at the origin, there exists $x\neq 0$ such that $$|\{y\in K: |< y,x> |\gr t\|<\cdot, x>\|_1\}|\ls\exp (-ct^2/\log^2(t+1))$$ for all $t\gr 1$, where $c>0$ is an absolute constant. The proof is based on the study of the $L_q$--centroid bodies of $K$. Analogous results hold true for general log-concave measures.