Estimates for moments of general measures on convex bodies

For $p\ge 1$, $n\in \mathbb{N}$, and an origin-symmetric convex body $K$ in $\mathbb{R}^n,$ let

$d_{\rm {ovr}}(K,L_p^n) = \inf \left \{ \Big (\frac {\vert D\vert}{\vert K\vert}\Big )^{1/n}: K \subseteq D,\ D\in L_p^n \right \}$

be the outer volume ratio distance from $K$ to the class $L_p^n$ of the unit balls of $n$-dimensional subspaces of $L_p.$ We prove that there exists an absolute constant $c>0$ such that

$\frac {c\sqrt {n}}{\sqrt {p\log \log n}}\le \sup _K d_{\rm {ovr}}(K,L_p^n)\le \sqrt {n}.$

This result follows from a new slicing inequality for arbitrary measures, in the spirit of the slicing problem of Bourgain. Namely, there exists an absolute constant $C>0$ so that for any $p\ge 1,$ any $n\in \mathbb{N}$, any compact set $K \subseteq \mathbb{R}^n$ of positive volume, and any Borel measurable function $f\ge 0$ on $K$,

$\int _K f(x)\,dx \, \le \, C \sqrt {p}\ d_{\rm ovr}(K,L_p^n)\ \vert K\vert^{1/n} \sup _{H} \int _{K\cap H} f(x)\,dx,$

where the supremum is taken over all affine hyperplanes $H$ in $\mathbb{R}^n$. Combining the above display with a recent counterexample for the slicing problem with arbitrary measures from the work of the second and third authors [J. Funct. Anal. 274 (2018), pp. 2089-2112], we get the lower estimate from the first display.

In turn, the second inequality follows from an estimate for the $p$-th absolute moments of the function $f$

$\min _{\xi \in S^{n-1}} \int _K \vert(x,\xi )\vert^p f(x)\ dx \, \le \, (Cp)^{p/2}\, d^p_{\rm {ovr}}(K,L_p^n)\ \vert K\vert^{p/n} \int _K f(x)\,dx.$

Finally, we prove a result of the Busemann-Petty type for these moments.