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Links 1 through 10 of 2207 by janus tagged preprints

In this paper we develop a numerical method for solving a class of optimization problems known as optimal location or quantization problems. The target energy can be written either in terms of atomic measures and the Wasserstein distance or in terms of weighted points and power diagrams (generalized Voronoi diagrams). The latter formulation is more suitable for computation. We show that critical points of the energy are centroidal power diagrams, which are generalizations of centroidal Voronoi tessellations, and that they can be approximated by a generalization of Lloyd's algorithm (Lloyd's algorithm is a common method for finding centroidal Voronoi tessellations). We prove that the algorithm is energy decreasing and prove a convergence theorem. Numerical experiments suggest that the algorithm converges linearly. We illustrate the algorithm in two and three dimensions using simple models of optimal location and crystallization. In particular, we test a conjecture about ...

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We provide a full quantitative version of the Gaussian isoperimetric inequality. Our estimate is independent of the dimension, sharp on the decay rate with respect to the asymmetry and with optimal dependence on the mass.

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We present Lipschitz continuity estimates for a class of convex functions with respect to Hormander vector fields. These results have been obtained in collaboration with M. Scienza.

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We present a new blow-up method that allows for establishing the first general formula to compute the perimeter measure with respect to the spherical Hausdorff measure in noncommutative nilpotent groups. Our techniques are new also in the classical Euclidean framework. When the distance is sub-Riemannian, we are lead to an unexpected relationship between the validity of a suitable area formula with respect to a distance and the profile of its corresponding unit ball.

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We introduce and study new invariants associated with Laplace type elliptic partial differential operators on manifolds. These invariants are constructed by using the off-diagonal heat kernel; they are not pure spectral invariants, that is, they depend not only on the eigenvalues but also on the corresponding eigenfunctions in a non-trivial way. We compute the first three low-order invariants explicitly.

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We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known examples except the totally geodesic ones and spherical caps are unstable.

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The deficit in the logarithmic Sobolev inequality for the Gaussian measure is considered and estimated by means of transport and information-theoretic distances.

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Inspired by ideas of R. Schatten in his celebrated monograph on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes E of a pointed metric space and a Banach space E as a certain linear subspace of the algebraic dual of Lipo(X,E^*). We prove that forms a dual pair. We prove that X\boxtimes E is linearly isomorphic to the linear space of all finite-rank continuous linear operators from (X^#,T) into E, where X^# denotes the space Lipo(X,K) and T is the topology of pointwise convergence of X^#. The concept of Lipschitz tensor product of elements of X^# and E^* yields the space X^#\boxast E^* as a certain linear subspace of the algebraic dual of X\boxtimes E.

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In this paper we disprove a conjecture stated in [4] on the equality of two notions of dimension for closed cones. Moreover, we answer in the negative to the following question, raised in the same paper. Given a compact family  of closed cones and a set S such that every blow-up of S at every point x∈S is contained in some element of , is it true that the dimension of S is smaller than or equal to the largest dimension of a vector space contained is some element of ?

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The Steiner tree problem can be stated in terms of finding a connected set of minimal length containing a given set of finitely many points. We show how to formulate it as a mass-minimization problem for 1-dimensional currents with coefficients in a suitable normed group. The representation used for these currents allows to state a calibration principle for this problem. We also exhibit calibrations in some examples.

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