On the uniqueness ofLp-Minkowski problems: The constant p-curvature case inR3

Abstract

Abstract. We study the C 4 smooth convex bodies K ⊂ R n+1 satisfying K(x) = u(x) 1−p , where x ∈ S n , K is the Gauss curvature of ∂K, u is the support function of K, and p is a constant. In the case of n = 2, either when p ∈ [−1, 0] or when p ∈ (0, 1) in addition to a pinching condition, we show that K must be the unit ball. This partially answers a conjecture of Lutwak, Yang, and Zhang about the uniqueness of the Lp-Minkowski problem in R 3 . Moreover, we give an explicit pinching constant depending only on p when p ∈ (0, 1).

A flow method for the dual Orlicz–Minkowski problem

“…The L p Minkowski problem has been extensively studied, see e.g. [5,7,8,15,17,28,32,36,39,40,41,47,48,49,50,52,54,67,68] and Schneider's book [59], and corresponding references therein. When p = 0, Eq.…”

The dual Minkowski problem for symmetric convex bodies

“…Alternate proofs were given by Hug-Lutwak-Yang-Zhang [31]. The case of p < 1 is still largely open (see Böröczky-Lutwak-Yang-Zhang [9], Huang-Liu-Xu [28], Jian-Lu-Wang [32], and Zhu [55,57]). For other recent progress on the L p -Minkowski problem, see Böröczky-Trinh [11] and Chen-Li-Zhu [12,13].…”

Continuity of the solution to the even logarithmic Minkowski problem in the plane

“…Besides, the centro-affine Minkowski problem (p = −n) and the logarithmic Minkowski problem (p = 0) are other two special cases of the L p Minkowski problem, see [4,7,11,22,36,37,48,50]. For the existence, uniqueness and regularity of the (normalized) L p Minkowski problem, one can see [9,18,21,24,26,27,29,49,52]. As an important application, the solutions to the L p Minkowski problem play a vital role in discovering some new (sharp) affine L p Sobolev inequalities, see [10, 15-17, 28, 43, 44].…”