---
title: Logarithmic Minkowski Problem
url: https://www.emergentmind.com/topics/logarithmic-minkowski-problem
type: topic
---

# Logarithmic Minkowski Problem

The Logarithmic Minkowski Problem is a central question in convex geometric analysis, generalizing the classical Minkowski problem by seeking to prescribe the cone-volume measure rather than the surface-area measure of a convex body. The analytic and geometric structures revealed by this problem integrate variational, PDE, and mass-transport techniques, and connect it deeply to the broader $L_p$-Minkowski theory.

## 1. Definition and Formulation

Given a finite Borel measure $\mu$ on the unit sphere $S^{n-1}$, the Logarithmic Minkowski Problem asks for necessary and sufficient conditions for the existence of a convex body $K \subset \mathbb{R}^n$ (with the origin in its interior) such that $\mu$ is the cone-volume measure $V_K$ of $K$. Specifically, for every Borel set $\omega \subset S^{n-1}$,
$$
V_K(\omega) = \int_{\nu_K^{-1}(\omega)} \langle x,\nu_K(x) \rangle \, d\mathcal{H}^{n-1}(x),
$$
where $\nu_K(x)$ denotes the outer unit normal at $x \in \partial K$, and $\mathcal{H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure.

The Logarithmic Minkowski Problem is a singular member ($p=0$) of the $L_p$-Minkowski hierarchy, with connections to variational principles and fully nonlinear spherical Monge–Ampère equations. When $K$ is origin-symmetric—equivalently, a unit ball of a finite-dimensional Banach space—$\mu$ must be an even measure, defining the even Logarithmic Minkowski Problem [2502.05430].

## 2. Key Notions and Analytic Framework

### Support and Surface Area Functions

- **Support function**: $h_K(u)=\max_{x\in K}\langle u, x\rangle$, $u\in S^{n-1}$. $h_K$ is 1-homogeneous, subadditive, and continuous.
- **Gauss map**: $\nu_K: \partial K \to S^{n-1}$, which pushes forward $\mathcal{H}^{n-1}$ to the classical surface-area measure $S_K$ via $S_K(\omega) = \mathcal{H}^{n-1}\{x \in \partial K : \nu_K(x) \in \omega\}$.

### Cone-Volume Measure

- For the polytope case with facet normals $u_i$ and associated face volumes $v_i$, $V_K = \sum_i v_i \delta_{u_i}$.
- For smooth $K$, $dV_K(u) = h_K(u)dS_K(u)$. This is the $L_0$-surface-area measure in the $L_p$-Minkowski hierarchy [2502.05430].

### Spherical Monge–Ampère Equation

If $\mu$ is absolutely continuous, $d\mu(u) = f(u) dH^{n-1}(u)$, seeking $K$ with $V_K=\mu$ leads to
$$
h\det(h_{ij} + h\delta_{ij}) = f(u), \quad u \in S^{n-1},
$$
where $h_{ij}$ are the second covariant derivatives of $h$ on $S^{n-1}$.

## 3. Existence: Variational and Inductive Methods

### Subspace Concentration Condition

For even measures, the necessary and sufficient solvability condition is a subspace-concentration:
> $\mu(W \cap S^{n-1}) \leq \frac{\dim W}{n}\,\mu(S^{n-1})$ for all $W \subset \mathbb{R}^n$, $0 < \dim W < n$,
with equality cases requiring $\mu$ supported on complementary subspaces [2502.05430].

This condition is sharp and, in the even case, characterizes precisely the cone-volume measures of symmetric convex bodies [2502.05430]. For discrete measures, analogous restrictions apply, with existence for polytopes whose facet normals satisfy the discrete concentration inequality [1409.7907].

### Variational Principle

Existence is proved by minimizing the functional
$$
\Phi_\mu(K) = \int_{S^{n-1}} \log h_K(u)\, d\mu(u),
$$
over origin-symmetric $K$ with volume matching $\mu(S^{n-1})$. Compactness and "blow-up" (degeneration) lemmas, combined with a first-variation computation, yield existence and all minimizers give $V_K = \mu$ [2502.05430].

### Symmetrization and Inductive Step

Necessity of the subspace-concentration is verified via symmetrization: one shows that if $\mu$ is a cone-volume measure, this inequality holds (with potential equality structure captured via direct sum decompositions). Induction over dimension applies when equality occurs, reducing to lower-dimensional problems [2502.05430].

## 4. Uniqueness, Refinements, and Open Problems

### Even Logarithmic Minkowski Conjecture

Lutwak's conjecture posits uniqueness for symmetric bodies under the subspace-concentration condition. Proven in dimension $n=2$ and for certain symmetric classes (e.g., reflection or unconditional symmetry), uniqueness near the Euclidean ball (for $h_K$ close to constant) follows from spectral-gap-based local analysis [2210.00194, 1811.10181].

In general, global uniqueness remains open, particularly for general even positive data and for non-symmetric measures [2210.00194, 2502.05430].

### Refined Necessary Conditions

Recent results provide sharper necessary conditions by supplementing subspace-concentration with additional inequalities. For any direction $u$,
$$
n(x(u)+y(u)) + (n+1)^{n-1} |x(u)-y(u)|^n \leq 1
$$
where $x(u)$, $y(u)$ are the normalized cone-volumes at $u$ and $-u$. This strictly refines the previous conditions and generalizes two-dimensional results to all $n \geq 3$ [2601.18167].

The question of whether these refinements yield sufficiency is open.

### Regularity

For the Monge–Ampère PDE associated to the log-problem, regularity results show: in $\mathbb{R}^3$, any solution with smooth, positive density is $C^{1,1}$, which is proven to be optimal for $p=0$ [2304.11617].

## 5. The Discrete and Planar Logarithmic Minkowski Problems

For discrete measures (support on finite sets of normals), the problem reduces to matching cone-volume vectors via a system of algebraic and geometric constraints. Existence and uniqueness are characterized in terms of subspace-concentration, but recent planar results uncover additional U-dependent linear inequalities which any admissible cone-volume vector must satisfy, sharpening the classical bounds and providing a polytope description for feasibility [2601.13159].

For discrete-convex body settings in higher dimensions, general-position and non-hemisphere support conditions yield existence and uniqueness for both the volume and (generalized) capacity analogs [2111.07321, 1409.7907].

## 6. Extensions: Mass Transport, Gaussian and Cone Cases

### Mass-Transport and Variational Characterizations

Optimal transport provides new functionals (e.g., the Kantorovich cost with the cost $c(x,y)=\log(1/\langle x, y \rangle)$) whose minimizers solve the log-Minkowski problem for symmetric data, forging links to entropy-transport inequalities and to Kähler–Einstein theory [1807.07002]. The variational landscape includes transport-entropy functionals whose critical points realize prescribed cone-measure [1807.07002].

### Gaussian Logarithmic Minkowski Problem

In the Gaussian context, where Lebesgue measure is replaced by the standard Gaussian, the analogous problem arises for $C$-pseudo-cones. Existence results hold without any subspace-concentration restriction, but non-uniqueness is generic, due to the lack of homogeneity and translation-invariance in the Gaussian setting [2502.10039]. Evolution by suitable curvature flows can also produce solutions in the Gaussian context [2212.01822].

### Log-Minkowski Problem for $C$-Coconvex Sets

In the setting of $C$-coconvex sets (with respect to a pointed convex cone $C$), the log-Minkowski existence and uniqueness theory is complete. The paper [2204.00860] establishes both a log-Brunn–Minkowski inequality and (crucially) uniqueness of solutions, settling an open question of Schneider.

## 7. Special Cases and Examples

- **Planar Case ($n=2$):** The structure is comprehensively understood—uniqueness for origin-symmetric bodies, explicit necessary and sufficient conditions, and refinement via discrete inequalities [2502.05430, 2601.18167, 2601.13159].
- **Euclidean Ball:** For measures proportional to spherical Lebesgue measure, the solution is the unit ball [2502.05430].
- **$\ell_p^n$ unit balls:** Direct computation shows the associated cone measures satisfy strict subspace concentration, so the corresponding problem is uniquely and explicitly solvable [2502.05430].
- **Curvature-Pinched Bodies:** For $K$ whose anisotropic metric is within a factor $\gamma\leq n+1$ of the Euclidean metric, global uniqueness and the even log-Minkowski inequality hold [2307.16484].
- **Symmetry Classes:** Uniqueness is established for bodies invariant under $n$ hyperplane reflections or unconditional bodies [2002.12239, 2101.03395].

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**References**: The above account synthesizes results from [2502.05430], [2601.18167], [2210.00194], [1811.10181], [2304.11617], [2002.12239], [1807.07002], [2111.07321], [1409.7907], [2212.01822], [2502.10039], [2204.00860], [2601.13159], [2307.16484].

Source: https://www.emergentmind.com/topics/logarithmic-minkowski-problem