No abstract

The classical Loomis-Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide lower bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer's dual Loomis-Whitney inequality.

We investigate a Maclaurin inequality for vectors and its connection to an Aleksandrov-type inequality for parallelepipeds.

We provide a new quantitative version of Helly's theorem: there exists an absolute constant α > 1 with the following property: if {Pi : i ∈ I} is a finite family of convex bodies in R n with int i∈I Pi = ∅, then there exist z ∈ R n , s αn and i1, . . . is ∈ I such thatwhere c > 0 is an absolute constant. This directly gives a version of the "quantitative" diameter theorem of Bárány, Katchalski and Pach, with a polynomial dependence on the dimension. In the symmetric case the bound O(n 3/2 ) can be improved to O( √ n).

We provide new quantitative versions of Helly's theorem. For example, we show that for every family {Pi:i∈I} of closed half‐spaces in Rn such that P=⋂i∈IPi has positive volume, there exist s⩽αn and i1,…,is∈I such that right center left3ptthickmathspacetrueitalicvolMJX-TeXAtom-ORDn⁡(PMJX-TeXAtom-ORDiMJX-TeXAtom-ORD1∩⋯∩PMJX-TeXAtom-ORDiMJX-TeXAtom-ORDs)⩽(Cn) MJX-TeXAtom-ORDnitalicvolMJX-TeXAtom-ORDn⁡(P), where α,C>0 are absolute constants. These results complement and improve previous work of Bárány et al and Naszódi. Our method combines the work of Srivastava on approximate John's decompositions with few vectors, a new estimate on the corresponding constant in the Brascamp–Lieb inequality and an appropriate variant of Ball's proof of the reverse isoperimetric inequality.