Some Affine Invariants Revisited

Stancu, Alina

doi:10.1007/978-1-4614-6406-8_16

Cited by 4 publications

(3 citation statements)

References 41 publications

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“…In [74] Stancu introduces new equicentro-affine invariants, and she provides a geometric interpretation of the L Φ affine surface area introduced by Ludwig and Reitzner [50]. Further applications are given by Stancu in [75] in connection to the PaourisWerner invariant defined on convex bodies [59].…”

Section: Introductionmentioning

confidence: 99%

The planar Busemann-Petty centroid inequality and its stability

Ivaki

2015

Trans. Amer. Math. Soc.

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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.

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Section: Introductionmentioning

confidence: 99%

The planar Busemann-Petty centroid inequality and its stability

Ivaki

2015

Trans. Amer. Math. Soc.

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“…There are several important contributions of geometric flows to convex geometry, for example, a proof of the affine isoperimetric inequality by B. Andrews using the affine normal flow [1], obtaining the necessary and sufficient conditions for the existence of a solution to the discrete L 0 -Minkowski problem using crystalline curvature flow by A. Stancu [35,36,39] and independently by B. Andrews [3], and a proof of the p-affine isoperimetric inequality in the class of origin-symmetric convex bodies in R 2 using the affine normal [19]. See [37,38,40,41] for more applications of flows, in particular, a newly defined family of centro-affine p-flows and their applications to centro-affine differential geometry by A. Stancu [40,41].…”

Section: Introductionmentioning

confidence: 99%

On the Stability of the p-Affine Isoperimetric Inequality

Ivaki

2013

J Geom Anal

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Abstract. Employing the affine normal flow, we prove a stability version of the p-affine isoperimetric inequality for p ≥ 1 in R 2 in the class of origin-symmetric convex bodies. That is, if K is an origin-symmetric convex body in R 2 such that it has area π and its p-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, K is close to an ellipse in the Hausdorff distance.

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“…In [74] Stancu introduces new equicentro-affine invariants, and she provides a geometric interpretation of the L Φ affine surface area introduced by Ludwig and Reitzner [50]. Further applications are given by Stancu in [75] in connection to the Paouris-Werner invariant defined on convex bodies [59].…”

Section: Introductionmentioning

confidence: 99%

Centro-Affine Invariants for Smooth Convex Bodies

Stancu

2011

International Mathematics Research Notices

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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.2010 Mathematics Subject Classification. Primary 52A40, 53C44, 52A10; Secondary 35K55, 53A15.

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Some Affine Invariants Revisited

Cited by 4 publications

References 41 publications

The planar Busemann-Petty centroid inequality and its stability

The planar Busemann-Petty centroid inequality and its stability

On the Stability of the p-Affine Isoperimetric Inequality

Centro-Affine Invariants for Smooth Convex Bodies

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