arXiv subjects
Publications and source records attributed to Mohammad N. Ivaki.
Using polar convex bodies and the $C_0$-bounds from Guan and Ni \cite{PL}, we obtain a uniform lower bound on the Gauss curvature of the normalized solution of the Gauss curvature flow without using Chow's Harnack inequality \cite{Ch2}.
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 $1\leq p<\infty.$ 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 suitably normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo $SL(2).$ 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.
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 $SL(n+1)$.
We prove that the only compact convex ancient solutions of the planar affine normal flow are contracting ellipses.
We prove that the only compact, origin-symmetric, strictly convex ancient solutions of the planar $p$ centro-affine normal flows are contracting origin-centered ellipses.
We give a stability version of of the Blaschke-Santaló inequality in the plane.
We prove that the set of smooth, $π$-periodic, positive functions on the unit circle for which the $L_{-2}$ Minkowski problem is solvable is dense in the set of all smooth, $π$-periodic, positive functions on the unit circle with respect to the $L^{\infty}$ norm. Furthermore, we obtain a necessary condition on the solvability of the even $ L_{-2}$ Minkowski problem. At the end, we prove uniqueness of the solutions up to an affine linear transformation.
By means of dual convex bodies, we obtain regularity of solutions to the expanding Gauss curvature flows with homogeneity degrees $-p$, $0<p<1$. At the end, we remark that our method can also be used to obtain regularity of solutions to the shrinking Gauss curvature flows with homogeneity degrees less than one.
Employing the affine normal flow, we prove a stability version of the $p$-affine isoperimetric inequality for $p\geq1$ in $\mathbb{R}^2$ in the class of origin-symmetric convex bodies. That is, if $K$ is an origin-symmetric convex body in $\mathbb{R}^2$ 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.
A generalization of the affine-geometric Wirtinger inequality for curves to hypersurfaces is given.
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 $π$ 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 $π$, 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, $p\geq 1$, for smooth, centrally symmetric convex bodies in $\mathbb{R}^2$.