Steiner symmetrals and their distance from a ball (Q1424035)

For \(\varepsilon> 0\), let \(N(n,\varepsilon)\) be the minimum number of successive Steiner symmetrizations sufficient to transform any convex body in \(\mathbb{R}^n\), with volume equal to the volume of the unit ball \(B^n\), into a convex body whose Hausdorff distance from \(B^n\) is at most \(\varepsilon\). Complementing known upper estimates of this number, the authors first construct, for any positive integers \(n\geq 2\) and \(m\), an \(n\)-dimensional convex body with the property that any \(m\) successive Steiner symmetrizations do not decrease the distance of the transformed body from a ball. From this, they deduce that \[ N(n,\varepsilon)\geq {\log(\log(1/\varepsilon))\over\log 2} \,(1+ o(1))\qquad\text{as}\quad\varepsilon\to 0. \]