On iterations of Steiner symmetrizations (Q325954)

Let \(N\) be a positive integer and let \(u \in \mathbb{S}^{N-1} := \{ x \in \mathbb{R}^N : \| x \| = 1 \}\) denote the unit sphere. The \textit{Steiner symmetrization} in the direction \(u\) is a mapping \(S_u\) of measurable subsets of \(\mathbb{R}^N\) to measurable subsets of \(\mathbb{R}^N\), acting as follows. Let \(E \subseteq \mathbb{R}^N\) be measurable, and let \(l_u := \{ \lambda u : \lambda \in \mathbb{R} \}\) denote the line through the origin in direction \(u\). Let \(u^\perp\) denote the \((N-1)\)-dimensional subspace orthogonal to \(u\). For each \(x \in u^\perp\) define \(c(x)\) as follows. If \(E \cap ( x + l_u ) = \emptyset\), then \(c(x) := \emptyset\). Otherwise, \(c(x)\) is the closed line segment on \(x + l_u\) centred at \(x\) with length equal to the one-dimensional outer measure of \(E \cap ( x + l_u )\). Then \(S_u E\) is defined as the union over all \(x \in u^\perp\) of \(c(x)\). The main result of the paper is as follows. Suppose that \(U\) is a countable and dense subset of \(\mathbb{S}^{N-1}\). Then there exists an ordering \(\{ u_n \}\) of \(U\) such that the Steiner symmetrizations of any compact set \(K \subset \mathbb{R}^N\), taken successively in directions \(u_n\), converge in the Hausdorff distance to the ball \(K^*\) centred at the origin having the same volume as \(K\).