Hemispherical Minkowski Problem

• Hemispherical Minkowski Problem is a variant where prescribed measures are restricted to a hemisphere, requiring modified balance conditions and boundary treatments.
• It employs adapted Monge–Ampère equations and variational as well as discrete methods to construct constant curvature surfaces, reflectors, and capillary designs.
• Analytical and numerical approaches, including power diagrams and quasi-Newton methods, address challenges in existence, uniqueness, and regularity under singular data constraints.

The Hemispherical Minkowski Problem is a variant of the classical Minkowski problem in convex and differential geometry, in which data are prescribed not globally on the entire unit sphere but only on a hemisphere, or on the complement of a finite set of points. This paradigm appears naturally in the construction of constant curvature surfaces with singularities, reflector design in geometric optics, and the analysis of convex bodies or surfaces satisfying Monge–Ampère type equations with boundary or singularity constraints. The hemispherical context is characterized by either the restriction of curvature or measure data to a hemisphere of the sphere $\mathbb{S}^n$ or by the relaxation of the traditional balance (centering) condition necessary for existence and uniqueness in the classical Minkowski problem.

1. Classical and Hemispherical Minkowski Problems: General Framework

The classical Minkowski problem seeks a convex body $K \subset \mathbb{R}^n$ whose surface area measure $S(K, \cdot)$ is a prescribed Borel measure $\mu$ on the sphere $\mathbb{S}^{n-1}$, subject to the equilibrium condition

$\int_{\mathbb{S}^{n-1}} u\, d\mu(u) = 0,$

which prevents concentration of $\mu$ in any closed hemisphere. The support function $h: \mathbb{S}^{n-1} \to \mathbb{R}_+$ of $K$ must solve the fully nonlinear Monge–Ampère equation

$\det(D^2 h + hI) = f$

on $\mathbb{S}^{n-1}$, where $f$ is the density of $\mu$.

In the Hemispherical Minkowski Problem ("Hemisphere MP" Editor's term), the prescribed measure $\mu$ is supported only on a hemisphere or on the complement of a finite set of points of $\mathbb{S}^{n-1}$. The standard balance condition generally fails, so the problem formulation, solvability, and analytic tools must be adjusted. Alternative strategies include restricting the domain of the support function, imposing Neumann or other boundary conditions along the equator, or introducing modified compatibility conditions for the PDE. In many physical, geometric, or analytic scenarios—including generalized reflector problems, spacelike surfaces in Minkowski space, and the construction of constant Gauss curvature ("CGC") surfaces—this hemispherical setup is inherent (Alarcon et al., 2012, Huang et al., 8 Feb 2025, Castro et al., 2014).

2. Existence, Uniqueness, and Analytical Structure

The existence and uniqueness theory for the Hemispherical Minkowski Problem diverges from the classical theory due to the failure of global centering. For data supported outside a finite set $\{p_j\}_{j=1}^m \subset \mathbb{S}^2$, one considers singular measures of the form

$$\mu = \mu_{\mathbb{S}^2} + \sum_{j=1}^m a_j \delta_{p_j}$$

with $a_j > 0$, where $\mu_{\mathbb{S}^2}$ is the spherical Lebesgue measure and $\delta_{p_j}$ are Dirac measures (Alarcon et al., 2012). The equilibrium condition becomes a vector constraint:

$\sum_{j=1}^m a_j p_j = 0.$

Given appropriate data, the Minkowski problem admits a unique (up to translation) solution: a convex body $K$ with boundary consisting of a smooth open constant curvature surface $S_\mathrm{open}$ (e.g., with $K=1$), and $m$ planar compact discs $\overline{D_j}$ orthogonal to $p_j$. The regularity is $C^1$ and piecewise analytic, and the Gauss map of $S_\mathrm{open}$ is a diffeomorphism onto $\mathbb{S}^2 \setminus \{p_j\}$. The support function $h$ on the sphere (or its punctured domain) solves a Monge–Ampère equation with singularities:

$\det(\nabla^2 h + hI) = 1/\kappa,$

where $\kappa$ is the prescribed curvature, extended as a measure possibly with singular parts.

For domains such as the open disk (in the Klein model), the hemispherical or domain-restricted Minkowski problem can be reformulated as a Monge–Ampère equation

$\det D^2u(z) = \frac{1}{\psi(z)} (1 - |z|^2)^{-2}$

with appropriate Dirichlet or lower semicontinuous boundary data. Existence and uniqueness are obtained if the boundary data is bounded and $\psi$ remains strictly positive (Bonsante et al., 2015, Bonsante et al., 2018). The uniqueness can be shown using a comparison principle for the Monge–Ampère equation.

3. Methodologies: Variational, Discrete, and Computational Approaches

For the hemispherical variant, classical variational methods based on global volume or entropy functionals require localization. In discrete or polyhedral settings, the measure $\mu$ is often supported on finitely many directions. Algorithmic approaches, especially in geometric optics or far-field reflector problems, model reflectors as intersections of confocal paraboloids, yielding a cell decomposition on $\mathbb{S}^{n-1}$. The combinatorial and numerical challenge of the Hemispherical Minkowski Problem can be addressed by:

• Reformulating the cell decomposition using power diagrams intersected with the unit sphere, thus encoding the "hemispherical Minkowski cell structure" (Castro et al., 2014).
• Employing variational principles: maximizing or minimizing functionals defined on positive support functions or Wulff shapes (e.g., for Orlicz, $L^p$, or weighted Minkowski problems) (Chen et al., 2020, Langharst et al., 29 Jul 2024).
• Computing or approximating solutions to systems relating the areas or volumes of "cells" to prescribed data, often via quasi-Newton or topological methods.
• For PDE-based methods, solving fully nonlinear elliptic equations (Monge–Ampère form) with Dirichlet or Neumann conditions on a hemisphere or a punctured domain (Huang et al., 8 Feb 2025).

In the super-critical $L_p$-Minkowski regime ($p < -n-1$), topological arguments based on the homology of spaces of ellipsoids or hemispherical ellipsoids are employed to construct initial data or establish the existence of solutions (Guang et al., 2022).

4. Applications in Differential and Convex Geometry

The hemispherical Minkowski framework has enabled advances and explicit constructions in several areas:

• Constant Curvature Surfaces: New examples are obtained in $\mathbb{R}^3$ under the hemispherical Minkowski setup. The method constructs surfaces of constant Gauss curvature $K=1$ covering all of $\mathbb{S}^2$ minus a finite set, with planar discs attached at the omitted points, effectively "gluing" a CGC surface to planar boundary pieces (Alarcon et al., 2012).
• Harmonic and Minimal Lagrangian Maps: The Gauss map for the open part of such a surface yields a harmonic diffeomorphism between a circular domain and a punctured sphere, or a minimal Lagrangian map when the CGC surface is considered in Minkowski 3-space (Alarcon et al., 2012, Bonsante et al., 2018).
• Capillary and H-Surfaces: Outer parallel surfaces of the constructed CGC surfaces produce constant mean curvature H-surfaces meeting planar boundaries at specified contact angles, relevant for modeling capillary phenomena in containers (Alarcon et al., 2012).
• Generalized Reflector Problems: In geometric optics, the hemispherical Minkowski paradigm addresses reflector design with target intensities or prescribed energy distributions over directions confined to a hemisphere (Castro et al., 2014).
• Fully Nonlinear Hessian Equations: Solutions to Monge–Ampère type equations on punctured or hemispherical domains correspond bijectively to geometric data (such as the areas of omitted discs and their normals) (Alarcon et al., 2012).

The methodology generalizes broadly to Orlicz–Brunn–Minkowski theory, dual and $L^p$ Minkowski problems, and $p$-harmonic measure-prescribed Minkowski problems (Gardner et al., 2018, Akman et al., 2023, Chen et al., 2020, Langharst et al., 29 Jul 2024).

5. Structural and Analytic Challenges: Uniqueness, Regularity, and Balance Conditions

A defining challenge of the hemispherical problem is the absence of global centering. When $\mu$ is supported in a hemisphere, traditional Minkowski theory fails to guarantee uniqueness or even existence without further modifications. Alternative techniques include:

• Imposing boundary conditions (often of Neumann type) along the equator in hemispherical domains (Huang et al., 8 Feb 2025, Guang et al., 2022).
• Restricting the function space to positive support functions on the hemisphere, possibly adjusting normalization to counteract the lack of balance.
• Handling singular measures (e.g., Dirac masses) by verifying vector equilibrium conditions such as $\sum_j a_j p_j = 0$ (Alarcon et al., 2012).
• Investigating the regularity of solutions (with the appearance of non-removable singularities at the missing points or at the boundary of the hemisphere) (Alarcon et al., 2012, Bonsante et al., 2015).
• Employing function spaces characterized by Zygmund regularity or Thurston norm bounds for boundary data, reflecting geometric finiteness properties (Bonsante et al., 2015, Bonsante et al., 2018).

Uniqueness in the hemispherical setting often fails without additional geometric or analytic constraints. For weighted and Orlicz settings, existence and uniqueness are established under rotational invariance and non-concentration of the data measure on any great hemisphere, conditions which must be adapted or relaxed in the hemispherical case (Langharst et al., 29 Jul 2024).

6. Connections to Related Minkowski-Type Problems and Open Directions

The hemispherical Minkowski paradigm is intertwined with broader developments in convex geometric analysis and fully nonlinear PDEs:

• General Geometric Measures: The theory extends to prescribe cone–volume, Orlicz, $L^p$, and dual curvature measures localized on hemispheres, with associated Monge–Ampère-type PDEs accompanied by non-standard boundary conditions (Huang et al., 8 Feb 2025, Chen et al., 2020, Gardner et al., 2018).
• Discrete and Algorithmic Settings: The computational modeling of reflector and polyhedral problems via power diagrams or convex hulls intersects with discrete hemispherical Minkowski theory (Castro et al., 2014).
• Harmonic Analysis on the Sphere: Analytic techniques, such as the use of the spherical Radon transform, mass transport, or expansion in spherical harmonics, are adapted to treat localization and singular support phenomena arising in hemisphere-restricted Minkowski problems (Huang et al., 8 Feb 2025).
• Stability and Geometric Inequalities: The validity of Minkowski-type inequalities is sensitive to convexity and geometry in the hemispherical case. For instance, even arbitrarily small (in $W^{2,p}$ norm) non-convex perturbations of the round sphere can violate the Minkowski inequality with optimal constant, demonstrating the necessity of convexity in global geometric constraints (Chodosh et al., 2023).

A plausible implication is that further research will address uniqueness, regularity, and optimization for hemispherical Minkowski problems with strongly unbalanced or data-supported-on-partial-sphere measures, possibly through refined variational formulations or by leveraging insights from convex geometric analysis and fully nonlinear PDEs.

7. Summary Table: Key Aspects of the Hemispherical Minkowski Problem

Aspect	Classical MP	Hemispherical MP
Data support	Full sphere $\mathbb{S}^{n-1}$	Hemisphere or subset
Balance condition	$\int u\,d\mu(u)=0$	Fails; require alternate
PDE	Monge–Ampère on $\mathbb{S}^{n-1}$	On a hemisphere/subset, with BC
Uniqueness	Generally holds	May fail unless modified
Typical applications	Convex bodies, closed surfaces	Surfaces with singularities, reflectors, capillary surfaces, etc.

The Hemispherical Minkowski Problem is thus a central variant in convex geometric analysis and geometric PDE, distinguished by the localization (or omission) of data support, the breakdown of global centering, and the ensuing need for adapted analytic, geometric, and variational approaches. Its paper informs the construction of new geometric examples, advances algorithmic and applied design problems, and bridges convex geometry with nonlinear analysis and harmonic analysis.