Convex bodies with pinched Mahler volume under the centro-affine normal flows. (English) Zbl 1323.53077
Summary: 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 \(\mathcal {C}^\infty\) topology modulo \(\mathrm{SL}(n+1)\).
MSC:
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 52A05 | Convex sets without dimension restrictions (aspects of convex geometry) |
| 35K55 | Nonlinear parabolic equations |
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
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