In the discrete setting, the L 0 -Minkowski problem extends the question posed and answered by the classical Minkowski's existence theorem for polytopes. In particular, the planar extension, which we address in this paper, concerns the existence of a convex polygonal body which contains the origin, whose boundary sides have preassigned orientations and each triangle formed by the origin with two consecutive vertices is of prescribed area.

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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.

Abstract. We study the long time behavior of the volume preserving p-flow in R n+1 for 1 ≤ p < n+1 n−1. By extending Andrews' technique for the flow along the affine normal, we prove that every centrally symmetric solution to the volume preserving p-flow converges sequentially to the unit ball in the C ∞ topology, modulo the group of special linear transformations.

Abstract. Let K ⊂ R n+1 be a convex body of class C 2 with everywhere positive Gauss curvature. We show that there exists a positive number δ(K) such that for any, where K δ , K δ and K * stand for the convex floating body, the illumination body, and the polar of K, respectively. We derive a few consequences of these inequalities.