Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems

“…the books by R. Gardner [16] and R. Schneider [46] and also [30,33,35,52,53,54,58]. Recent developments include extensions to an Orlicz theory, e.g., [17,27,33,59], to a functional setting [12,13] and to the spherical and hyperbolic setting [5,6]. Applications of affine surface areas have been manifold.…”

Section: Introductionmentioning

confidence: 99%

Constrained convex bodies with extremal affine surface areas

Giladi

Huang

Schütt

et al. 2020

Journal of Functional Analysis

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Given a convex body K ⊆ R n and p ∈ R, we introduce and study the extremal inner and outer affine surface areaswhere asp(K ′ ) denotes the Lp-affine surface area of K ′ , and the supremum is taken over all convex subsets of K and the infimum over all convex compact subsets containing K. The convex body that realizes IS1(K) in dimension 2 was determined in [3] where it was also shown that this body is the limit shape of lattice polytopes in K. In higher dimensions no results are known about the extremal bodies. We use a thin shell estimate of [22] and the Löwner ellipsoid to give asymptotic estimates on the size of ISp(K) and osp (K). Surprisingly, it turns out that both quantities are proportional to a power of volume. *

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“…For the convex body K and the hypersurface M = ∂K, consider the dual body K * , by [22] we know r * = 1 u , and…”

mentioning

confidence: 99%

A class of anisotropic expanding curvature flows

Sheng¹,

Yi²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in Euclidean R n+1 with speed u α σ β k firstly, where u is support function of the hypersurface, α, β ∈ R 1 , and β > 0, σ k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1 ≤ k ≤ n. For α ≤ 1 − kβ, β > 1 k we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for α ≤ 1 − kβ, β > 1 k , we prove that the flow with the speed f u α σ β k exists for all time and converges smoothly after normalisation to a soliton which is a solution of f u α−1 σ β k = c provided that f is a smooth positive function on S n and satisfies that (∇ i ∇ j f 2010 Mathematics Subject Classification. 35K96, 53C44.

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Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems

Cited by 231 publications

References 69 publications

A flow method for the dual Orlicz–Minkowski problem

A flow method for the dual Orlicz–Minkowski problem

Constrained convex bodies with extremal affine surface areas

A class of anisotropic expanding curvature flows

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