The planar Busemann-Petty centroid inequality and its stability

Ivaki, Mohammad N.

doi:10.1090/tran/6503

Cited by 12 publications

(10 citation statements)

References 76 publications

(89 reference statements)

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“…For p > 2, ϕ ≡ 1, the C 1 convergence follows from the work of Chow-Gulliver [9] (up-to showing convexity is preserved). For p = −n − 1, k = n, ϕ ≡ 1, the flow was studied in [12,13] and for p > −n − 1, k = n, ϕ ≡ 1 in [3]. Acknowledgment M.I.…”

Section: An Expanding Flowmentioning

confidence: 99%

Deforming a hypersurface by principal radii of curvature and support function

Ivaki

2018

Calc. Var.

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We study the motion of smooth, closed, strictly convex hypersurfaces in R n+1 expanding in the direction of their normal vector field with speed depending on the kth elementary symmetric polynomial of the principal radii of curvature σ k and support function h. A homothetic self-similar solution to the flow that we will consider in this paper, if exists, is a solution of the well-known Lp-Christoffel-Minkowski problem ϕh 1−p σ k = c. Here ϕ is a preassigned positive smooth function defined on the unit sphere, and c is a positive constant. For 1 ≤ k ≤ n − 1, p ≥ k + 1, assuming the spherical hessian of ϕ where p ∈ R, E i is the ith elementary symmetric polynomial of principal curvatures for 0 ≤ i ≤ n normalized so that E i (1, . . . , 1) = 1 (and E 0 ≡ 1), ν(·, t) is the outer unit normal vector of M t := F (M n , t) and ϕ is a positive smooth function defined on the unit sphere S n .Assuming M t is strictly convex, its support function as a function on the unit sphere is given byWriteḡ and∇ for the standard round metric and the Levi-Civita connection of S n . Recall that the principal radii of curvature are the eigenvalues of the matrix r ij :=∇ i∇j h +ḡ ij h

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Section: An Expanding Flowmentioning

confidence: 99%

Deforming a hypersurface by principal radii of curvature and support function

Ivaki

2018

Calc. Var.

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“…For some choices of f and ϕ(s) = s 1− p , the flow (2.1) becomes homogeneous and was considered in [3,8,16,24,[34][35][36][37][38]44,53,55,57]. However, when it comes to non-homogeneous flows, the literature on geometric flows is not very rich and there are few works in this direction; e.g., [11,[16][17][18]42,45,50].…”

Section: Curvature Flowsmentioning

confidence: 99%

Orlicz–Minkowski flows

2021

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We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if they exist, solve the regular Orlicz–Minkowski problems. As an application, we obtain old and new existence results for the regular even Orlicz–Minkowski problems; the corresponding $$L_p$$ L p version is the even $$L_p$$ L p -Minkowski problem for $$p>-n-1$$ p > - n - 1 . Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz–Minkowski problems; the $$L_p$$ L p versions are the even $$L_p$$ L p -Minkowski problem for $$p>0$$ p > 0 and the $$L_p$$ L p -Minkowski problem for $$p>1$$ p > 1 . In the final section, we use a curvature flow with no global term to solve a class of $$L_p$$ L p -Christoffel–Minkowski type problems.

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“…To obtain a lower bound on the centro-affine curvature, we may first establish a Harnack estimate. Although, Harnack inequality could be avoided, we present it here for future applications, such as stability of some inequalities [18,19]. In dimension two, the Harnack estimate for the p-flow was proved in [20] with an application to classification of compact ancient solutions.…”

Section: Upper and Lower Bounds On The Centro-affine Curvaturementioning

confidence: 99%

“…We continue with the following observation on obtaining lower bounds on the speed which first appeared in Smoczyk [33] in his study of flow of star-shaped hypersurfaces by the mean curvature, and has been used in quite a few papers since then [8,9,19].…”

Section: Upper and Lower Bounds On The Centro-affine Curvaturementioning

confidence: 99%

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Convex bodies with pinched Mahler volume under the centro-affine normal flows

Ivaki

2014

Calc. Var.

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We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke-Santaló inequality, we obtain regularity of the solutions provided that initial convex bodies have almost maximum Mahler volume. We prove that suitably rescaled solutions converge sequentially to the unit ball in the C ∞ topology modulo SL(n + 1).2010 Mathematics Subject Classification. Primary 53C44, 52A05; Secondary 35K55. Key words and phrases. Geometric flows, Centro-affine normal flow, Centro-affine curvature, Blaschke-Santaló inequality, p-affine isoperimetric inequality.It was proved in [34] that if {K t } [0,T ) evolves by the p-flow, then {K * t } [0,T ) is a solution of the following evolution equation, the expanding p-flow (alternatively called the dual p-flow):

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The planar Busemann-Petty centroid inequality and its stability

Cited by 12 publications

References 76 publications

Deforming a hypersurface by principal radii of curvature and support function

Deforming a hypersurface by principal radii of curvature and support function

Orlicz–Minkowski flows

Convex bodies with pinched Mahler volume under the centro-affine normal flows

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