The planar Busemann-Petty centroid inequality and its stability. (English) Zbl 1335.52022
Summary: In [Int. Math. Res. Not. 2012, No. 10, 2289–2320 (2012; Zbl 1250.52009)], A. Stancu introduced a family of centro-affine normal flows, \( p\)-flows, for \( 1\leq p<\infty\). Here we investigate the asymptotic behavior of the planar \( p\)-flow for \( p=\infty \), in the class of smooth, origin-symmetric convex bodies. First, we prove that the \( \infty \)-flow evolves appropriately normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo \(\mathrm{SL}(2)\). Second, using the \( \infty \)-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 \( \mathcal {C}^{\infty }\) topology.
MSC:
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 52A10 | Convex sets in \(2\) dimensions (including convex curves) |
| 35K55 | Nonlinear parabolic equations |
| 53A15 | Affine differential geometry |
Keywords:
affine support function; Banach-Mazur distance; centro-affine normal flow; centroid body; geometric evolution equation; Hausdorff metric; Busemann-Petty centroid inequality; stabilityCitations:
Zbl 1250.52009References:
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