On the uniqueness of Lp-Minkowski problems: The constant p-curvature case in R3

By Yong Huang, Jiakun Liu, Lu Xu. 2015

93 citations recorded by the source corpus.

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 L p -Minkowski problem in R 3 . Moreover, we give an explicit pinching constant depending only on p when p ∈ ( 0 , 1 ) .

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/99f5d12b969c52a5f3ed07b7769a67269600f8c9

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In Applied Mathematics

1. Yiming Zhao
2. Yong Huang
3. K. Boroczky
4. Deane Yang
5. Gaoyong Zhang
6. Erwin Lutwak
7. Qi-Rui Li
8. Deping Ye

Citations in the corpus

• Local Lp-Brunn-Minkowski inequalities for p<1 (2017, background): “sitive curvature, and [63] for existence and uniqueness in the even planar problem when µ is assumed discrete). Various other partial results pertaining to the uniqueness question are known (see e.g. [31, 52, 67, 12] and the references therein). Without assuming evenness of µ, both existence and uni”
• The dual Minkowski problem for symmetric convex bodies (2017, background): “The case of p < 1 is still largely open (see Böröczky-LutwakYang-Zhang [9], Huang-Liu-Xu [28], Jian-Lu-Wang [32], and Zhu [55, 57]).”
• A flow method for the dual Orlicz–Minkowski problem (2020, background): “3) is the Lp Minkowski problem, which includes the logarithmic Minkowski problem (p = 0) and the centroaffine Minkowski problem (p = −n). The Lp 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],”
• On the L dual Minkowski problem (2018, result): “Important recent results in this direction include Andrews [2], Böröczky, Lutwak, Yang & Zhang [9], Choi & Daskalopoulos [18], Huang, Liu & Xu [31], and Jian, Lu & Wang [36].”
• Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems (2016, unknown): “, [18], [20], [43], [44], [54], [55], and [84].”