Volume preserving centro-affine normal flows

Ivaki, Mohammad N.; Stancu, Alina

doi:10.4310/cag.2013.v21.n3.a9

Cited by 33 publications

(31 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To obtain the positive lower bound for the principal curvatures of M t , we will study an expanding flow by Gauss curvature for the dual hypersurface of M t . This technique was previously used in [12,30,31,32]. Expanding flows by Gauss curvature have been studied in [23,24,38,40,41].…”

Section: By (23) and (42)mentioning

confidence: 99%

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Sheng

Wang

2019

J. Eur. Math. Soc.

View full text Add to dashboard Cite

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space R n+1 with speed f r α K, where K is the Gauss curvature, r is the distance from the hypersurface to the origin, and f is a positive and smooth function. If α ≥ n + 1, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if f ≡ 1. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang [29], for the case q < 0. If α < n+1, corresponding to the case q > 0, we also establish the same results for even function f and originsymmetric initial condition, but for non-symmetric f , counterexample is given for the above smooth convergence.2010 Mathematics Subject Classification. 35K96, 53C44.

show abstract

Section: By (23) and (42)mentioning

confidence: 99%

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Sheng

Wang

2019

J. Eur. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…In particular, the affine isoperimetric inequality implies the Blaschke-Santaló inequality and it proved to be the key ingredient in the solution of many problems, see e.g. 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].…”

Section: Introductionmentioning

confidence: 99%

Constrained convex bodies with extremal affine surface areas

Giladi

Huang

Schütt

et al. 2020

Journal of Functional Analysis

View full text Add to dashboard Cite

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. *

show abstract

“…In [41,44] it was shown that the evolution equation of polar bodies combined with Tso's trick and SalkowskiKaltenbach-Hug identity (see [37, Theorem 2.8]) provide a useful tool for obtaining regularity of solutions to a class of geometric flows. Proof of Lemma 4.1 is omitted because of its similarity to the proof of [41, Theorem 2.2].…”

Section: Long Time Behaviormentioning

confidence: 99%

“…The long time behavior of the flow in R n with K 0 ∈ F n e was investigated in [38,44,76]. It was proved that for 1 < p < n n−2 the volume-preserving p-flow, which keeps the volume of the evolving bodies xed and equal to the volume of the unit ball, evolves each body in F n e to the unit ball in C ∞ , modulo SL(n).…”

Section: Introductionmentioning

confidence: 99%

The planar Busemann-Petty centroid inequality and its stability

Ivaki

2015

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, p-flow, for 1 ≤ p < ∞. Here we investigate the asymptotic behavior of the planar p-flow for p = ∞ in the class of smooth, origin-symmetric convex bodies. First, we prove that the ∞-flow evolves suitably normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo SL(2). Second, using the ∞-flow and a Harnack estimate for this flow, we prove a stability version of the planar Busemann-Petty centroid inequality in the Banach-Mazur distance. Third, we prove that the convergence of normalized solutions in the Hausdorff metric can be improved to convergence in the C ∞ topology.

show abstract

Volume preserving centro-affine normal flows

Cited by 33 publications

References 11 publications

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Constrained convex bodies with extremal affine surface areas

The planar Busemann-Petty centroid inequality and its stability

Contact Info

Product

Resources

About