[Paper Review] The Lp Minkowski problem for polytopes for negative p

Guangxian Zhu|arXiv (Cornell University)|Feb 25, 2016

Point processes and geometric inequalities31 references3 citations

TL;DR

This paper establishes the existence of solutions to the $L_p$ Minkowski problem for polytopes when $p < 0$, including the critical case $p = -n$ (centro-affine Minkowski problem). Using a variational approach and perturbation argument on support functions, the author proves that for any discrete measure with positive weights on unit normals not concentrated on a hemisphere and satisfying subspace concentration conditions, there exists a polytope whose $L_p$ surface area measure matches the given measure.

ABSTRACT

Existence of solutions to the Lp Minkowski problem is proved for all p less than 0. For the cirtical case of p=-n, which is known as the centro-affine Minkowski problem, this paper contains the main result in [71] as a special case.

Motivation & Objective

• To resolve the $L_p$ Minkowski problem for polytopes in the previously unexplored regime of negative $p$, particularly for $p < 0$.
• To extend the known existence results for $p > 1$ to the case $p < 0$, where the $L_p$ Minkowski inequality is not established and the problem is significantly more challenging.
• To address the critical case $p = -n$, known as the centro-affine Minkowski problem, which is invariant under centro-affine transformations and central to affine isoperimetric theory.
• To establish sufficient conditions on discrete measures (positive weights on unit normals) for the existence of a polytope realizing the measure as its $L_p$ surface area measure.
• To prove that the solution exists under the condition that the normals are not concentrated on any hemisphere and their intersections with any subspace lie within a closed hemisphere.

Proposed method

• Formulates the $L_p$ Minkowski problem for polytopes as a variational problem minimizing a functional $\Phi_P(\xi)$ related to the $L_p$ surface area measure.
• Applies a perturbation technique by modifying a polytope $P$ with outer normals $u_k$ to $P_\delta$, shifting facets inward by $\delta$ to create a new polytope $P_\delta$.
• Uses a scaling factor $\lambda_\delta = V(P_\delta)^{-1/n}$ to normalize the volume of the perturbed polytope to 1, ensuring the origin is the centroid.
• Analyzes the behavior of the functional $\Phi_{\lambda P_\delta}(\xi(\lambda P_\delta))$ under perturbation, showing it strictly decreases under certain conditions.
• Employs a contradiction argument: assuming a minimizer $P$ exists with $\Phi_P(o) < \Phi_{P_0}(o)$ for some $P_0$, then constructs a perturbed polytope $P_0$ with strictly smaller functional value, contradicting minimality.
• Relies on the convexity of $t^p$ for $t > 0$ and $p < 0$, and asymptotic analysis of the scaling factor $\lambda_\delta$ as $\delta \to 0^+$ to show $B(\delta) > 0$, implying a decrease in the functional.

Experimental results

Research questions

• RQ1 Does the $L_p$ Minkowski problem for polytopes have a solution when $p < 0$, particularly in the critical case $p = -n$?
• RQ2 Can the existence of a polytope be guaranteed for a discrete measure $\sum \alpha_k \delta_{u_k}$ on the sphere when $p < 0$, under suitable geometric constraints on the normals?
• RQ3 What conditions on the configuration of outer unit normals $u_k$ ensure the existence of a solution to the $L_p$ Minkowski problem for $p < 0$?
• RQ4 How does the functional $\Phi_P(\xi)$ behave under small perturbations of the polytope, and can this be used to prove minimality and existence?
• RQ5 Is the centro-affine Minkowski problem solvable for discrete measures on polytopes when the normals are in general position and not concentrated on a hemisphere?

Key findings

• The $L_p$ Minkowski problem for polytopes has a solution for all $p < 0$, extending the known results from $p > 1$ to the negative range.
• For $p = -n$, the result includes the centro-affine Minkowski problem as a special case, confirming the existence of solutions under the same conditions.
• The solution exists if the unit normals $u_1, \dots, u_N$ are not contained in any open hemisphere and their intersections with any subspace lie within a closed hemisphere.
• The existence is proven via a contradiction argument using a perturbation of the support function and a volume normalization, showing that any minimizer must satisfy the required measure condition.
• The functional $\Phi_P(\xi)$ strictly decreases under a specific perturbation when $p < 0$, which contradicts the assumption of a minimizer unless the solution exists.
• The proof relies on the convexity of $t^p$ for $t > 0$ and $p < 0$, and the asymptotic expansion of the volume normalization factor $\lambda_\delta$ as $\delta \to 0^+$, which ensures the perturbation yields a strictly smaller functional value.