The Volume Helly Theorem in the plane, colorful version

Abstract

We prove a colorful volume Helly theorem for convex sets in ${\mathbb{R}}^2$: There is a constant $V>0$ such that if ${\mathfrak{F}}_1,{\mathfrak{F}}_2,{\mathfrak{F}}_3,{\mathfrak{F}}_4$ are finite families of convex sets in ${\mathbb{R}}^2$ and if $|{\bigcap }_1^4{F}_i|\ge V$ for every transversal $F_i\in {\mathfrak{F}}_i,(i=1,\dots ,4)$, then $|\bigcap {\mathfrak{F}}_i|\ge 1$ for some $i$. Here $|A|$ is the Lebesgue measure of $A\subset {\mathbb{R}}^d$. The main ingredient is the following theorem. Let $Q_1,\dots ,Q_4\subset {\mathbb{R}}^2$ be convex quadrilaterals of area at most $1$, where of course each $Q_i$ is the intersection of 4 halfplanes. Then for every $Q_i$ there is one of these halfplanes $H_i$, say, such that $|{\bigcap }_1^4{H}_i|\le 4096$.