Isoperimetry of waists and concentration of maps
Let \(S^n\) be the unit \(n\)-sphere. A measurable partition \(\Pi\) of \(S^n\) into convex subsets \(S=S_\pi,\;\pi\in \Pi\), is called convex if it equals the limit of consecutive refinements of finite partitions into convex subsets. Let \(\mu\) be a Borel probability measure on \(S^n\), \(\mathcal{M}_k^\prime\) the space of weak limits \(\mu^\prime\) of the restrictions \((\mu| U_i)/\mu(U_i)\) for all sequences of convex open \(U_i\)'s that Hausdorff converge to compact convex subsets \(S\subset S^n\) of codimension (exactly!) \(k\) and assume that \(S=\text{supp}\, \mu^\prime\). Let \(c^k_\bullet:\mathcal{M}_k^\prime\to S^n\) be a continuous mapping such that \(c^k_\bullet(\mu^\prime)\in S=\text{supp}\, \mu^\prime\) for all \(\mu^\prime\) (e.g., the center of mass of \(\mu^\prime\)). If \(\Pi\) is a convex partition of \(S^n\) into \(S_\pi\)'s with \(\text{codim}\, S_\pi\geq k\), then \(\mu\) induces the canonical system of probability measures \(\mu_\pi\) on almost all \(S_\pi\) and these \(\mu_\pi\) are convexly derived from \(\mu\) for almost all \(\pi\in \Pi\) [see \textit{A. Giannopoulos} and \textit{V. Milman}, Euclidean structure in finite dimensional spaces, in: Handbook of the Geometry of Banach Spaces, Vol. 1, 707--779 (2001; Zbl 1009.46004)], and thus -- via the ``center'' map \(c^k_\bullet\) -- the ``central'' subset \(C_\pi\) is well defined in \(S_\pi\) for almost all \(\pi\). If \(f:S^n\to \mathbb{R}^k\) is a continuous map -- under the above assumptions -- the author proves that there exist a value \(z\in \mathbb{R}^k\) and a convex partition \(\Pi\) of \(S^n\) into \(S_\pi\)'s of codimension \(\geq k\), such that the level \(Y_z:=f^{-1}(z)\subset S^n\) meets \(C_\pi\) for almost all \(\pi\in \Pi\). In particular, there exists a point \(z\in \mathbb{R}^k\) such that the spherical \(n\)-volumes of the \(\varepsilon\)-neighbours of the level \(Y_z\), denoted \(Y_z+\varepsilon\subset S^n\), satisfy \[ \text{Vol}(Y_z+\varepsilon)\geq \text{Vol}(S^{n-k}+\varepsilon)\tag \((\star)_{S^n}\) \] for all \(\varepsilon >0\), where \(S^{n-k}\subset S^n\) denotes an equatorial \((n-k)\)-sphere. The inequality \((\star)_{S^n}\) for \(\varepsilon\to 0\) shows that the Minkowski \((n-k)\)-volume of \(Y_z\) is \(\geq \) than that of the equatorial sphere \(S^{n-k}\subset S^n\). Special attention is devoted to the case \(k=1\). Thus, for example, if \(k=1\) and the Lévy mean of \(f\) for \(z\in \mathbb{R}\) is considered, where the level \(Y_z\subset S^n\) divides \(S^n\) into equal halves, the author shows that \(\star_{S^n}\) follows from the spherical isoperimetric inequality. If \(k=n\) and \(\text{card}\, Y_z\leq 2,\;z\in\mathbb{R}^n\), then \(\star_{S^n}\) applied to \(\varepsilon=\pi/2\) amounts to the Borsuk-Ulam theorem: some level \(Y_z\) of \(f:S^n\to \mathbb{R}^n\) equals a pair of opposite points. If \(c^k_\bullet\) and \(C_\pi\) are as above and \(k<n\), then the conclusion of the partition theorem holds true for every Borsuk-Ulam family of (codimension \(k\) ``cycles'') subsets in \(S^n\). Examples and applications of this last result are given. The author provides an elementary solution to the Gehring linking problem [see the author, J. Differ. Geom. 18, 1--147 (1983; Zbl 0515.53037), and references therein]. Parametric partitions and related results are also discussed. Many open questions are brought to light.
scientific article; zbMATH DE number 1971828