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On the uniqueness of Lp-Minkowski problems: The constant p-curvature case in R^3

Journal Article


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Abstract


  • We study the C4 smooth convex bodies K ⊂Rn+1 satisfying K(x) =u(x)1−p, where x ∈Sn, 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 R3. Moreover, we give an explicit pinching constant depending only on p when p ∈(0, 1).

Authors


  •   Huang, Yong (external author)
  •   Liu, Jiakun
  •   Xu, Lu (external author)

Publication Date


  • 2015

Citation


  • Huang, Y., Liu, J. & Xu, L. (2015). On the uniqueness of Lp-Minkowski problems: The constant p-curvature case in R^3. Advances in Mathematics, 281 906-927.

Scopus Eid


  • 2-s2.0-84931261582

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=5267&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/4246

Has Global Citation Frequency


Number Of Pages


  • 21

Start Page


  • 906

End Page


  • 927

Volume


  • 281

Place Of Publication


  • United Kingdom

Abstract


  • We study the C4 smooth convex bodies K ⊂Rn+1 satisfying K(x) =u(x)1−p, where x ∈Sn, 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 R3. Moreover, we give an explicit pinching constant depending only on p when p ∈(0, 1).

Authors


  •   Huang, Yong (external author)
  •   Liu, Jiakun
  •   Xu, Lu (external author)

Publication Date


  • 2015

Citation


  • Huang, Y., Liu, J. & Xu, L. (2015). On the uniqueness of Lp-Minkowski problems: The constant p-curvature case in R^3. Advances in Mathematics, 281 906-927.

Scopus Eid


  • 2-s2.0-84931261582

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=5267&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/4246

Has Global Citation Frequency


Number Of Pages


  • 21

Start Page


  • 906

End Page


  • 927

Volume


  • 281

Place Of Publication


  • United Kingdom