On the average volume of subsets in Euclidean \(d\)-space (Q1178025)

The authors prove that for given positive numbers \(\mu_ 0\), \(D\) and a positive integer \(d\), there exists a constant \(c=c(\mu_ 0,D,d)\) satisfying the following property: If \({\mathcal H}\) is a finite family of connected open subsets of \(E^ d\) with the volume at least \(\mu_ 0\) and the diameter at most \(D\), the average volume of sets which are the union of some elements of \({\mathcal H}\), is at least \(c\cdot\mu(\cup{\mathcal H})\), where \(\mu(\cup{\mathcal H})\) is the volume of the union of all elements of \({\mathcal H}\). In the case of \({\mathcal H}\) being a finite system of unit intervals on a line \(c=1/3\).