English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

For any origin-symmetric convex body $K$ in $\mathbb{R}^n$ in isotropicposition, we obtain the bound: \[ M^*(K) \leq C \sqrt{n} \log(n)^2 L_K ~, \]where $M^*(K)$ denotes (half) the mean-width of $K$, $L_K$ is the isotropicconstant of $K$, and $C>0$ is a universal constant. This improves the previousbest-known estimate $M^*(K) \leq C n^{3/4} L_K$. Up to the power of the$\log(n)$ term and the $L_K$ one, the improved bound is best possible, andimplies that the isotropic position is (up to the $L_K$ term) an almost$2$-regular $M$-position. The bound extends to any arbitrary position,depending on a certain weighted average of the eigenvalues of the covariancematrix. Furthermore, the bound applies to the mean-width of $L_p$-centroidbodies, extending a sharp upper bound of Paouris for $1 \leq p \leq \sqrt{n}$to an almost-sharp bound for an arbitrary $p \geq \sqrt{n}$. The question ofwhether it is possible to remove the $L_K$ term from the new bound isessentially equivalent to the Slicing Problem, to within logarithmic factors in$n$.

On the mean-width of isotropic convex bodies and their associated $L_p$-centroid bodies