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The approximation of a circle with a fine square grid distorts the perimeter by a factor of $tfrac{4}{pi}$. We prove that this factor is the same on average for approximations of any curve with any Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.
Let $X_1,ldots,X_n$ be i.i.d. random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$ f_{d,beta} (x) = text{const} cdot (1-|x|^2)^{beta}, quad |x|leq 1, quad text{(the beta case)} $$ an
We study stability of the spectral gap and observable diameter for metricmeasure spaces satisfying the RCD(1, $infty$) condition. We show that if such a space has an almost maximal spectral gap, then it almost contains a Gaussian component, and the L
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
The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the surface areas of
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