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Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

Alexander I. Bobenko, Ivan Izmestiev (2008)

Annales de l’institut Fourier

We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of generalized convex...

On Hadwiger's problem on inner parallel bodies

Eugenia Saorín (2009)

Banach Center Publications

We consider the problem of classifying the convex bodies in the 3-dimensional space depending on the differentiability of their associated quermassintegrals with respect to the one-parameter-depending family given by the inner/outer parallel bodies. It turns out that this problem is closely related to some behavior of the roots of the 3-dimensional Steiner polynomial.

Operations between sets in geometry

Richard J. Gardner, Daniel Hug, Wolfgang Weil (2013)

Journal of the European Mathematical Society

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in n -dimensional Euclidean space n . It is proved that if n 2 , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, G L ( n ) covariant, and associative if and only if it is L p addition for some 1 p . It is also demonstrated that if n 2 ,...

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