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The \(L_p\) Minkowski problem for polytopes for \(0 < p < 1\). (English) Zbl 1335.52023

The general \(L_p\) Minkowski-problem, a cornerstone of modern convex geometry, asks for a characterization of the \(L_p\) surface area measure of convex bodies among the finite Borel measures on the sphere. These measures were introduced by E. Lutwak [J. Differ. Geom. 38, No. 1, 131–150 (1993; Zbl 0788.52007)].
The case \(p=1\) corresponds to the classical Minkowski problem, solved by Minkowski, Aleksandrov, Fenchelamp, Jessen; and the particular interesting limit case \(p=0\) is the logarithmic Minkowski problem. Whereas for \(p>1\), the problem is mainly solved (see, e.g. [D. Hug et al., Discrete Comput. Geom. 33, No. 4, 699–715 (2005; Zbl 1078.52008)]), the range \(0\leq p<1\) is still less well understood.
The paper under review contains the surprising solution of the \(L_p\) Minkowski-problem for \(0<p<1\) and for discrete measures, i.e., for polytopes. It is shown that a finite discrete measure on the sphere is the \(L_p\)-surface area measure of a polytope if and only if the support of the measure is not concentrated on a closed hemisphere.
It is interesting to note that for \(p=0,1\) there are additional conditions which a measure has to fulfill in order to be a \(L_p\) surface area measure (see, e.g., [K. Böröczky jun. et al., J. Am. Math. Soc. 26, No. 3, 831–852 (2013; Zbl 1272.52012)]).

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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