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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We explore some inequalities in convex geometry restricted to the class ofzonoids. We show the equivalence, in the class of zonoids, between a localAlexandrov-Fenchel inequality, a local Loomis-Whitney inequality, thelog-submodularity of volume, and the Dembo-Cover-Thomas conjecture on themonotonicity of the ratio of volume to the surface area. In addition to theseequivalences, we confirm these conjectures in ${\mathbb R}^3$ and we establishan improved inequality in ${\mathbb R^2}$. Along the way, we give a negativeanswer to a question of Adam Marcus regarding the roots of the Steinerpolynomial of zonoids. We also investigate analogous questions in the$L_p$-Brunn-Minkowski theory, and in particular, we confirm all of the aboveconjectures in the case $p=2$, in any dimension.

On the volume of the Minkowski sum of zonoids