On the number of solutions to the discrete two-dimensional L0-Minkowski problem

On the Lp Minkowski Problem for Polytopes

“…During the past decade various elements of the L p Brunn-Minkowski theory have attracted increased attention (see e.g. [3], [4], [5], [8], [9] [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [22], [24], [25], [26], [27], [28], [29]). …”

“…A normalized version (discussed below) of the problem was proposed and completely solved for p > 1 and even data in [19]. For d = 2, the important case p = 0 of the discrete-data L p Minkowski problem was dealt with by Stancu [26], [27].…”

Necessary subspace concentration conditions for the even dual Minkowski problem

“…The discrete, planar, even case of the logarithmic Minkowski problem, i.e., with respect to origin-symmetric convex polygons, was completely solved by Stancu [49,50], and later Zhu [55] as well as Böröczky, Hegedűs and Zhu [6] settled (in particular) the case when K is a polytope whose outer normals are in general position.…”

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.…”