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Register a new userWaist of maps measured via Urysohn width
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We discuss various questions of the following kind: for a continuous map $X to Y$ from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The $d$-width measures how well a space can be approximated by a $d$-dimensional complex. The results of this paper include the following. 1) Any piecewise linear map $f: [0,1]^{m+2} to Y^m$ from the unit euclidean $(m+2)$-cube to an $m$-polyhedron must have a fiber of $1$-width at least $frac{1}{2beta m +m^2 + m + 1}$, where $beta = sup_y text{ rk } H_1(f^{-1}(y))$ measures the topological complexity of the map. 2) There exists a piecewise smooth map $X^{3m+1} to mathbb{R}^m$, with $X$ a riemannian $(3m+1)$-manifold of large $3m$-width, and with all fibers being topological $(2m+1)$-balls of arbitrarily small $(m+1)$-width.
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The notion of the Urysohn $d$-width measures to what extent a metric space can be approximated by a $d$-dimensional simplicial complex. We investigate how local Urysohn width bounds on a riemannian manifold affect its global width. We bound the $1$-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of $n$-manifolds of considerable $(n-1)$-width in which all unit balls have arbitrarily small $1$-width. We also give examples of topologically simple manifolds that are locally nearly low-dimensional.
We solve the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for the Hilbert space in the context of the Urysohn universal metric space. This is achieved by solving a purely combinatorial problem involving a family of countable homogeneous metric spaces with finitely many distances.
We study the Gromov waist in the sense of $t$-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromovs original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no $t$-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromovs pancake argument to produce some estimates of $t$-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. For readers convenience, in one appendix of this paper we provide a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures. In the other appendix, we provide a comparison of different variations of Gromovs pancake method.
Let $K in R^d$ be a convex body, and assume that $L$ is a randomly rotated and shifted integer lattice. Let $K_L$ be the convex hull of the (random) points $K cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The asymptotic order of the mean width difference $W(l K)-W((l K)_L)$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $l to infty$.
Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the $ell$-norm of convex bodies whose Lowner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschlager verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Lowner ellipsoid is the Euclidean unit ball. Here we prove stronger stabili
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