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- W2071279438 abstract "We consider two types of $p$-centro affine flows on smooth, centrally symmetric, closed convex planar curves, $p$-contracting, respectively, $p$-expanding. Here $p$ is an arbitrary real number greater than 1. We show that, under any $p$-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the origin. Furthermore, the normalized curves with enclosed area $pi$ converge, in the Hausdorff metric, to the unit circle modulo SL(2). As a $p$-expanding flow is, in a certain way, dual to a contracting one, we prove that, under any $p$-expanding flow, curves expand to infinity in finite time, while the only homothetic solutions of the flow are ellipses centered at the origin. If the curves are normalized as to enclose constant area $pi$, they display the same asymptotic behavior as the first type flow and converge, in the Hausdorff metric, and up to SL(2) transformations, to the unit circle. At the end, we present a new proof of $p$-affine isoperimetric inequality, $pgeq 1$, for smooth, centrally symmetric convex bodies in $mathbb{R}^2$." @default.
- W2071279438 created "2016-06-24" @default.
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- W2071279438 date "2012-05-29" @default.
- W2071279438 modified "2023-09-27" @default.
- W2071279438 title "Centro-affine curvature flows on centrally symmetric convex curves" @default.
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