On General Concavity Extensions of Grünbaum Type Inequalities

Abstract

Given a strictly increasing continuous function $\phi :\mathbb {R}_{\ge 0} \longrightarrow \mathbb {R}\cup \{-\infty \}$ with $\lim _{t\rightarrow \infty }\phi (t) = \infty$, a function $f:[a,b] \longrightarrow \mathbb {R}_{\ge 0}$ is said to be $\phi$-concave if $\phi \circ f$ is concave. When $\phi (t) = t^p$, $p>0$, this notion is that of p-concavity whereas for $\phi (t) = \log (t)$ it leads to the so-called log-concavity. In this work, we show that if the cross-sections volume function of a compact set $K\subset \mathbb {R}^n$ (of positive volume) w.r.t. some hyperplane H passing through its centroid is $\phi$-concave, then one can find a sharp lower bound for the ratio $\textrm{vol}(K^{-})/\textrm{vol}(K)$, where $K^{-}$ denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. Moreover, other related results for $\phi$-concave functions (and involving the centroid) are shown.