22,861 research outputs found
Existence and uniqueness of optimal maps on Alexandrov spaces
The purpose of this paper is to show that in a finite dimensional metric
space with Alexandrov's curvature bounded below, Monge's transport problem for
the quadratic cost admits a unique solution
Data depth and floating body
Little known relations of the renown concept of the halfspace depth for
multivariate data with notions from convex and affine geometry are discussed.
Halfspace depth may be regarded as a measure of symmetry for random vectors. As
such, the depth stands as a generalization of a measure of symmetry for convex
sets, well studied in geometry. Under a mild assumption, the upper level sets
of the halfspace depth coincide with the convex floating bodies used in the
definition of the affine surface area for convex bodies in Euclidean spaces.
These connections enable us to partially resolve some persistent open problems
regarding theoretical properties of the depth
Combinatorial optimization over two random point sets
We analyze combinatorial optimization problems over a pair of random point
sets of equal cardinal. Typical examples include the matching of minimal
length, the traveling salesperson tour constrained to alternate between points
of each set, or the connected bipartite r-regular graph of minimal length. As
the cardinal of the sets goes to infinity, we investigate the convergence of
such bipartite functionals.Comment: 34 page
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