Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$

Karoly J. Boroczky, Shibing Chen, Weiru Liu, Christos Saroglou

TL;DR

The paper proves a $C^0$-estimate for the $L_p$ $q$-th dual Minkowski problem on $S^{2}$ in dimension $3$ for $p\in[0,1)$ and $q>2+p$, assuming the dual curvature density is bounded above and below. The argument combines Monge–Ampère theory on the sphere with a John ellipsoid decomposition, establishing a fundamental relation $r_1(K)r_2(K)r_3(K)\approx r_3(K)^{3-q+p}$ and reducing the problem to showing $r_1(K)\gtrsim r_3(K)$. A contradiction framework rules out two geometric configurations (Case I and Case II) by detailed-volume and curvature estimates, yielding uniform bounds on the size and diameter of $K$, and thus the $C^0$ estimate and a lower bound on the volume. The results also imply a near-isotropy uniqueness result when the density is Hölder-close to constant and $q$ is close to $3$, contributing to the understanding of existence and uniqueness in the $L_p$ dual Minkowski theory in dimension three. Overall, the work strengthens the linkage between convex geometric measures and fully nonlinear Monge–Ampère equations in low dimensions, with concrete geometric consequences such as diameter control and volume lower bounds for solutions.

Abstract

We prove the $C^0$ estimate for the $L_p$ $q$th dual Minkowski problem on $S^2$ under fairly general conditions; namely, when $p$ lies in [0,1) and $q>2+p$, and the $L_p$ $q$th dual curvarture is bounded and bounded away from zero. We note that it is known that the analogous $C^0$ estimate does not hold if $p<-1$ and $q=3$. As a corollary of our $C^0$ estimate, we deduce the uniqueness of the solution of the near isotropic $q$th $L_p$ dual Minkowski problem on $S^2$ if $q$ is close to 3 and the $q$th $L_p$ dual curvature is Holder close to be the constant one function.

Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$

TL;DR

The paper proves a -estimate for the -th dual Minkowski problem on in dimension for and , assuming the dual curvature density is bounded above and below. The argument combines Monge–Ampère theory on the sphere with a John ellipsoid decomposition, establishing a fundamental relation and reducing the problem to showing . A contradiction framework rules out two geometric configurations (Case I and Case II) by detailed-volume and curvature estimates, yielding uniform bounds on the size and diameter of , and thus the estimate and a lower bound on the volume. The results also imply a near-isotropy uniqueness result when the density is Hölder-close to constant and is close to , contributing to the understanding of existence and uniqueness in the dual Minkowski theory in dimension three. Overall, the work strengthens the linkage between convex geometric measures and fully nonlinear Monge–Ampère equations in low dimensions, with concrete geometric consequences such as diameter control and volume lower bounds for solutions.

Abstract

We prove the estimate for the th dual Minkowski problem on under fairly general conditions; namely, when lies in [0,1) and , and the th dual curvarture is bounded and bounded away from zero. We note that it is known that the analogous estimate does not hold if and . As a corollary of our estimate, we deduce the uniqueness of the solution of the near isotropic th dual Minkowski problem on if is close to 3 and the th dual curvature is Holder close to be the constant one function.

Paper Structure

This paper contains 8 sections, 14 theorems, 183 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Some basic notation and notions
  3. Regularity if the dual Minkowski curvature is bounded and bounded away from zero
  4. Basic estimate for Theorem \ref{['n3-ppos--C0-estimate']}
  5. Proof of Proposition \ref{['good-inequality-prop']}, and in turn of Theorem \ref{['n3-ppos--C0-estimate']}
  6. Lemmas needed for the proof of Proposition \ref{['good-inequality-prop']} under the assumption as in Case I ($r_1 \leq r_2\ll r_3$)
  7. Lemmas needed for the proof of Proposition \ref{['good-inequality-prop']} under the assumption as in Case II ($r_1 \ll r_2 \approx r_3$)
  8. Completing the argument for Proposition \ref{['good-inequality-prop']} and Theorem \ref{['n3-ppos--C0-estimate']}

Key Result

THEOREM 1.2

For $p\in[0,1)$, $q>2+p$ and $\lambda>1$, there exists a constant $C=C(\lambda,p,q)>1$ such that if holds on $S^{2}$ for a convex body $K\in \mathcal{K}^3_o$, then

Figures (6)

  • Figure 1: lemma \ref{['q3 main']}
  • Figure 2: lemma \ref{['q3 case I subcase i lemma']}
  • Figure 3: lemma \ref{['q3 case I subcase ii.1 lemma']}
  • Figure 4: lemma \ref{['q3 case I subcaseii.2 lemma']}
  • Figure 5: lemma \ref{['q3 case II subcase i lemma']}
  • ...and 1 more figures

Theorems & Definitions (24)

  • THEOREM 1.2
  • COROLLARY 1.3
  • THEOREM 3.1: Caffarelli
  • LEMMA 3.2
  • LEMMA 4.1
  • proof
  • COROLLARY 4.2
  • proof
  • PROPOSITION 4.3
  • LEMMA 5.1
  • ...and 14 more