The necessary condition for the discrete L0-Minkowski problem in $${\mathbb{R}}^{2}$$

Abstract

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

On the Orlicz Minkowski Problem for Polytopes

“…The solution of the L p Minkowski problem for polytopes for all p > 1 was given by Chou and Wang [8], while an alternate approach to this problem was presented by Hug et al [20]. Other approaches towards the L p Minkowski problem have also been extensively studied over the last years (see, e.g., [6,9,17,19,22,[51][52][53][54]). Despite impressive success in this direction, not all problems concerning the L p Minkowski problem are completely solved.…”

Log-Minkowski inequalities for the L p $L_{p}$ -mixed quermassintegrals

“…The log-Minkowski inequality belongs to log-Minkowski theory. For more research on log-Minkowski theory, we may refer to [13][14][15][16][17][18][19][20][21][22].…”

The planar Busemann-Petty centroid inequality and its stability

“…Within the last few years, a substantial amount of research was devoted to investigate applications of geometric flows to different areas of mathematics. In particular, there are several major contributions of geometric flows to convex geometry: a proof of the affine isoperimetric inequality by Andrews using the affine normal flow [4], obtaining the necessary and sufficient conditions for the existence of a solution to the discrete L 0 -Minkowski problem using crystalline curvature flow by Stancu [70,71,73] and independently by Andrews [8], an application of the affine normal flow to the regularity of minimizers of Mahler volume by Stancu [72], and obtaining quermassintegral inequalities for k-convex star-shaped domains using a family of expanding flows [33]. To state our stability result, we recall the Banach-Mazur distance.…”