Logarithmic Minkowski Problem

• The Logarithmic Minkowski Problem is a central concept in convex geometric analysis that generalizes the classical Minkowski problem by prescribing the cone-volume measure of convex bodies.
• It employs analytic, variational, and mass-transport methods to establish existence and uniqueness conditions, including critical subspace concentration requirements for even measures.
• Practical applications include solving spherical Monge–Ampère equations and extending the theory to Gaussian and C-coconvex settings, shedding light on both smooth and discrete cases.

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 (Böröczky et al., 8 Feb 2025).

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 (Böröczky et al., 8 Feb 2025).

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 (Böröczky et al., 8 Feb 2025).

This condition is sharp and, in the even case, characterizes precisely the cone-volume measures of symmetric convex bodies (Böröczky et al., 8 Feb 2025). For discrete measures, analogous restrictions apply, with existence for polytopes whose facet normals satisfy the discrete concentration inequality (Boroczky et al., 2014).

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 $V_K = \mu$ (Böröczky et al., 8 Feb 2025).

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 (Böröczky et al., 8 Feb 2025).

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 (Boroczky, 2022, Chen et al., 2018).

In general, global uniqueness remains open, particularly for general even positive data and for non-symmetric measures (Boroczky, 2022, Böröczky et al., 8 Feb 2025).

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 $n \geq 3$ (Lai et al., 26 Jan 2026).

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 $p=0$ (Choi et al., 2023).

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 (Baumbach, 19 Jan 2026).

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 (Kim et al., 2021, Boroczky et al., 2014).

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 (Kolesnikov, 2018). The variational landscape includes transport-entropy functionals whose critical points realize prescribed cone-measure (Kolesnikov, 2018).

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 (Shan et al., 14 Feb 2025). Evolution by suitable curvature flows can also produce solutions in the Gaussian context (Sheng et al., 2022).

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 (Yang et al., 2022) 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 (Böröczky et al., 8 Feb 2025, Lai et al., 26 Jan 2026, Baumbach, 19 Jan 2026).
• Euclidean Ball: For measures proportional to spherical Lebesgue measure, the solution is the unit ball (Böröczky et al., 8 Feb 2025).
• $\ell_p^n$ unit balls: Direct computation shows the associated cone measures satisfy strict subspace concentration, so the corresponding problem is uniquely and explicitly solvable (Böröczky et al., 8 Feb 2025).
• 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 (Ivaki et al., 2023).
• Symmetry Classes: Uniqueness is established for bodies invariant under $n$ hyperplane reflections or unconditional bodies (Böröczky et al., 2020, Boroczky et al., 2021).

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References: The above account synthesizes results from (Böröczky et al., 8 Feb 2025, Lai et al., 26 Jan 2026, Boroczky, 2022, Chen et al., 2018, Choi et al., 2023, Böröczky et al., 2020, Kolesnikov, 2018, Kim et al., 2021, Boroczky et al., 2014, Sheng et al., 2022, Shan et al., 14 Feb 2025, Yang et al., 2022, Baumbach, 19 Jan 2026, Ivaki et al., 2023).