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.

No abstract

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.

We prove that the sufficiency condition employed to show the existence and, in certain cases the uniqueness, of solutions to the discrete, planar L 0 -Minkowski problem with data containing, at least, a pair of opposite vectors is also a necessary condition. (2000): 28A75, 52A10. Mathematics Subject ClassificationThe classical Minkowski problem deals with the existence, uniqueness, regularity and stability of closed, convex hypersurfaces whose Gauss curvature, viewed as a function on the unit sphere, is preassigned. For an atomic measure on the unit sphere, the question concerns the existence and uniqueness of polytopes with facets of fixed normal directions and fixed surface areas. In the planar setting, the Minkowski problem consists of a sufficient and necessary condition for the existence of a convex polygon whose sides have preassigned lengths and orientations:an ordered set of directions in S 1 , not all in a half-disk, and let L = {l 1 , ..., l N } be an ordered set of strictly positive numbers. There exists a convex N-gon whose i-th side has outer normal → u i and, respectively, length l i if and only ifThis is the simplest and the trivial case of Minkowski's problem as the aforementioned condition represents simply the closure of a polygonal line with the desired properties. One should note that the convex polygon so obtained is unique up to translation. See [16] for a detailed discussion on the full extent of Minkowski's problem.Due to Lutwak [10], a significantly more difficult question is whether a measure on the unit sphere S n can be realized as the L p -surface area measure of a convex body, where p = n is some fixed real number. If so, is this body unique? Lutwak showed within the Brunn-Minkowski-Firey theory that the classical problem, corresponding to p = 1, generalizes naturally to the L p -Minkowski problem stated above. In [10] a solution to the even L p -Minkowski problem in R n+1 was given for all p ≥ 1 except for p = n when it was shown that no solution is possible. The problem is called even if the measure takes equal values on opposite directions of S n . It is conjectured that in the even case the convex body, if it exists, is centrally symmetric, [10].162

Let K be a convex body with boundary of class C 4 . We prove that, for a sufficiently small δ, the floating body K δ is homothetic to K if and only if K is an ellipsoid. The proof relies on properties of the affine curvature flow.

No abstract