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Uniqueness of Lp Minkowski problem in the supercritical range

Shi-Zhong Du

Abstract

The uniqueness of the $L_p$-Minkowski problem has been a long standing problem in convex geometry. In the groundbreaking paper by Brendle-Choi-Daskalopoulos (Acta Math, {\bf219}, 2017), a full uniqueness result was shown for the subcritical exponents $p\in(-n-1,1]$. In the supercritical range, the uniqueness problem is much more complicated, even on the planar case $n=1$. One of the famous results was shown by Andrews in (J. Amer. Math. Soc., {\bf16}, 2003), where he established that the uniqueness holds in the range $p\in(-7,-2)$ and fails to hold for the other supercritical exponents $p\in(-\infty,-7)$. In this paper, we study the same uniqueness problem in the full supercritical range $p\in(-2n-5,-n-1)$ for all higher dimensional cases $n\geq2$. We will prove that for $p\in(-2n-5,-n-1)$, the unique strongly symmetric solution is given by the unit sphere $§^n$. The uniqueness range $(-2n-5,-n-1)$ is optimal due to our recent preprint (arXiv: 2104.07426), where non-spherical strongly symmetric solutions have been constructed for all $p\in(-\infty,-2n-5)$. When considering general solutions which may not be symmetric, the uniqueness set $Γ$ of $p$ for which the uniqueness holds, is shown to be both relatively open and closed in the full interval $(-2n-5,-n-1)$.

Uniqueness of Lp Minkowski problem in the supercritical range

Abstract

The uniqueness of the -Minkowski problem has been a long standing problem in convex geometry. In the groundbreaking paper by Brendle-Choi-Daskalopoulos (Acta Math, {\bf219}, 2017), a full uniqueness result was shown for the subcritical exponents . In the supercritical range, the uniqueness problem is much more complicated, even on the planar case . One of the famous results was shown by Andrews in (J. Amer. Math. Soc., {\bf16}, 2003), where he established that the uniqueness holds in the range and fails to hold for the other supercritical exponents . In this paper, we study the same uniqueness problem in the full supercritical range for all higher dimensional cases . We will prove that for , the unique strongly symmetric solution is given by the unit sphere . The uniqueness range is optimal due to our recent preprint (arXiv: 2104.07426), where non-spherical strongly symmetric solutions have been constructed for all . When considering general solutions which may not be symmetric, the uniqueness set of for which the uniqueness holds, is shown to be both relatively open and closed in the full interval .

Paper Structure

This paper contains 19 sections, 48 theorems, 248 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $p$-Blaschke-Santaló functional inequality with remainder
  3. The $1$st and $2$nd variations of the $p$-Blaschke-Santaló functional
  4. Combined Inequality \ref{['e1.3']} implies reverse $p$-Blaschke-Santaló functional inequality
  5. Strong symmetric solutions: Theorem \ref{['t1.2']}
  6. Orthonormal lemmas for strong symmetric functions
  7. Uniqueness of strongly symmetric solutions in a slightly supercritical range $p\in(-n-1-\sigma_n,-n-1)$
  8. Infinitesimal Generator Lemma for solution sequence
  9. Mapping Representation for approximation sequence
  10. Prolongation formula and infinitesimal generator field
  11. Infinitesimal Generator Lemma
  12. A-priori estimates and openness of the uniqueness set
  13. Ellipsoid lemmas and a-priori estimates I
  14. A-priori estimates II and proof of Theorem \ref{['t1.5']}
  15. Kernel Property and closedness of $\Gamma_\sigma$
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $\sigma>0$, suppose that there exists some constant $\Theta^*_{\sigma}>0$ such that holds for all $p\in(-n-1-\sigma,-n-1)$ and the combined quotient Then the solution of e1.2 is unique for $p\in (-n-1-\sigma',-n-1)$, where $Z_K^\perp$ is the orthonormal component of $Z_K$ defined in e2.10, and $\lambda_3(K)$ is the third eigenvalue of the linearized eigenvalue problem e2.1.

Figures (1)

  • Figure 1: A regular dodecahedron in good position (source: Wikipedia)

Theorems & Definitions (95)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 85 more