2018/11/17 by Kateryna Tatarko, Tatarko, Kateryna, Elisabeth M. Werner +1 · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #Affine transformation #Combinatorics #Connection (principal bundle) #Convex body #Convex hull #Differential Geometry (math.DG) #Diffusion and Search Dynamics #Dual (grammatical number) #FOS: Mathematics #Geometry #Mathematical analysis #Mathematics #Minimal surface #Minkowski space #Morphological variations and asymmetry #Point processes and geometric inequalities #Pure mathematics #Regular polygon #Steiner tree problem #Surface (topology) #math.DG
paper · pdf · doi:10.48550/arxiv.1811.07225
published in arXiv (Cornell University) (Cornell University)
arxiv created 2018/11/17 · arxiv updated 2018/11/20
We prove an analogue of the classical Steiner formula for the Lp affine surface area of a Minkowski outer parallel body for any real parameters p. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature. They have the intrinsic volumes of the classical Brunn Minkowski theory and the dual quermassintegrals of the dual Brunn Minkowski theory as well as special cases. Properties of these new quantities are investigated, a connection to information theory among them. A Steiner formula for the s-th mixed Lp affine surface area of a Minkowski outer parallel body for any real parameters p and~s is also given.