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 Tools annual

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 monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We show \begin{align*} \frac{ \int_{E \cap \theta^+} f(x) dx }{ \int_E f(x)dx }  \geq \left(\frac{k \gamma+1}{(n+1) \gamma+1}\right)^{\frac{k\gamma+1}{\gamma}} \end{align*} for all $k$-dimensional subspaces$E\subset\mathbb{R}^n$, $\theta\in E\cap S^{n-1}$, and all $\gamma$-concavefunctions $f:\mathbb{R}^n\rightarrow [0,\infty)$ with $\gamma >0$, $0<\int_{\mathbb{R}^n} f(x)\, dx <\infty$, and $\int_{\mathbb{R}^n} x f(x)\, dx$at the origin $o\in\mathbb{R}^n$. Here, $\theta^+ := \lbrace x\, : \, \langlex,\theta\rangle \geq 0 \rbrace$. As a consequence of this result, we get thefollowing generalization of Gr\"unbaum's inequality: \begin{align*} \frac{\mbox{vol}_k(K\cap E\cap\theta^+) }{ \mbox{vol}_k(K\cap E) } \geq \left(\frac{k}{n+1} \right)^k \end{align*} for all convex bodies$K\subset\mathbb{R}^n$ with centroid at the origin, $k$-dimensional subspaces$E\subset\mathbb{R}^n$, and $\theta\in E\cap S^{n-1}$. The lower bounds in bothof our inequalities are the best possible, and we discuss the equalityconditions.

Gr\"unbaum's inequality for sections