Horo-Convexity in Non-Euclidean Spaces

• Horo-convexity is defined as domains or functions in non-Euclidean spaces that mimic Euclidean convexity using horospheres, horoballs, and Busemann functions.
• It enforces strict curvature bounds and smooth boundary regularity, enabling robust geometric analysis and stable inverse curvature flows.
• The concept bridges classical convexity with modern methods in PDE analysis and optimization on Hadamard manifolds, enhancing algorithmic performance.

Horo-convexity is an extension of the classical notion of convexity to non-Euclidean settings, built on the geometry of horospheres and Busemann functions. In hyperbolic space, on Hadamard manifolds, and on the sphere, horo-convexity captures those domains or functions whose interaction with horospheres or horoballs replicates central dual (supporting) features of Euclidean convexity, leading to powerful geometric, analytic, and optimization-theoretic consequences.

1. Definitions and Characterizations

Horo-convexity is defined in several geometric contexts, each leveraging horospheres, horoballs, or analogous constructs:

• In Hyperbolic Space $\mathbb{H}^n$:

A smooth bounded domain $\Omega \subset \mathbb{H}^n$ is horo-convex if for every boundary point $p \in \partial \Omega$, there exists a horosphere $S$ passing through $p$ such that $\Omega$ lies entirely on the convex side of $S$. Formally, if $b_\xi$ is a Busemann function associated to a point at infinity $\xi$, then for some $c$, $\Omega \subset \{b_\xi \leq c\}$ with $p \in \{b_\xi = c\}$. Analytically, this is equivalent to requiring all principal curvatures $\kappa_i \geq 1$ at every boundary point; for hypersurfaces, horo-convexity corresponds to the second fundamental form satisfying $II \geq I$ (Wei et al., 15 Oct 2025).

• On Hadamard Manifolds:

A function $f$ on a Hadamard manifold $M$ is h-convex if, at each point, there exists a scaled Busemann function $B_{y,v}$ such that $f(x) - f(y) \geq B_{y,v}(x)$ for all $x$. The sublevel sets of $f$ are intersections of horoballs, and this "outer" description is essential for algorithmic convexity in negatively curved geometries (Criscitiello et al., 22 May 2025).

• Unit Disk ($\mathbb{D}$)—Planar Models:

A domain $G \subset \mathbb{D}$ is horocyclically convex if for any $w \in \mathbb{D}\setminus G$, there exists a horodisk $H$ containing $w$ and disjoint from $G$. Every boundary point admits such a supporting horodisk (Arango et al., 31 Jul 2024).

• Unit Sphere ($S^{n+1}$), Northern Hemisphere:

Horo-convexity is defined for hypersurfaces as a pointwise principal curvature bound: at every $p$, $\kappa_\alpha \geq \frac{1-u}{\phi'(r)}$, where $u$ is the support function and $\phi'(r)$ is the derivative of the spherical warping function. Equality characterizes spherical horospheres (spheres tangent to the equator) (Pan et al., 14 Dec 2025).

2. Geometric and Analytic Structure

Horo-convexity closely parallels, but strengthens, ordinary geodesic convexity in several crucial aspects:

• Support and Curvature:

In hyperbolic or spherical contexts, horo-convexity mandates that domains or hypersurfaces are uniformly "bent" with curvature no less than that of a horosphere (or spherical horosphere analogue) at every boundary point. This requirement is strictly stronger than geodesic convexity, as it is possible to construct geodesically convex sets that fail to meet the stringent horospherical curvature lower bound (Wei et al., 15 Oct 2025, Pan et al., 14 Dec 2025).

• Intersections and Projections:

The intersection of horospheres or horoballs with horo-convex domains yields Euclidean-convex shapes in the corresponding horospherical coordinates. Horo-convexity is preserved under stereographic and radial projections to $\mathbb{R}^{n+1}$ from the sphere, resulting in convex images with principal curvatures uniformly bounded below (Pan et al., 14 Dec 2025).

• Regularity and Flow:

The boundaries of horo-convex domains are smooth (at least $C^\infty$ where specified) and admit smooth deformations under horospherical inverse curvature flows, which preserve horo-convexity and drive the domain boundary to the unique model object (sphere or horosphere) exponentially fast (Pan et al., 14 Dec 2025, Wei et al., 15 Oct 2025).

3. Horo-Convexity and Optimization

On Hadamard manifolds, horo-convexity enables a duality and envelope theory analogous to the Euclidean setting, which is unavailable for geodesic convexity:

• Functions and Supporting Structures:

Horo-convex (h-convex) functions admit supporting Busemann functions at every point, with the function itself recoverable as the pointwise supremum of such supports (envelope representation). Epigraphs of h-convex functions are intersections of horoballs in the manifold plus one dimension (Criscitiello et al., 22 May 2025).

• Algorithmic Impact:

Gradient and subgradient algorithms for minimizing (or maximizing) h-convex functions on Hadamard manifolds achieve oracle complexity matching Euclidean bounds—curvature independence is restored, as the critical supporting objects are horoballs, not geodesic half-spaces. Accelerated methods (Nesterov momentum) and Fréchet mean computations also transfer with the same rates. In negatively curved spaces ($\mathbb{H}^n$), localizations can even surpass the Euclidean rate for sublevel set identification (Criscitiello et al., 22 May 2025).

• Examples:

Scaled Busemann functions, distance functions, Fréchet mean, various functionals over symmetric positive definite matrices (e.g., Tyler's M-estimator), and key problems in robust and structured optimization (Horn's problem) are all h-convex (Criscitiello et al., 22 May 2025).

4. Connections to PDE, Analysis, and Metric Geometry

Horo-convexity arises naturally in the study of differential equations, functional inequalities, and metric comparison principles:

• Super Log-Concavity in Hyperbolic Domains:

The principal analytic result for horo-convex domains in hyperbolic space is that the first Dirichlet eigenfunction of the Laplacian is "super log-concave" when the domain has sufficiently small diameter. Specifically, for such domains, the matrix $-( \log v )_{ij} - |\nabla \log v|\,\delta_{ij}$ is positive semi-definite throughout, extending Euclidean concavity theory to hyperbolic space. If the domain is merely convex or has too large a diameter, log-concavity fails (Wei et al., 15 Oct 2025).

• Curvature Prescription and A Priori Estimates:

Horo-convexity permits solving nonlinear PDEs such as the prescribed shifted Gauss curvature equation for star-shaped hypersurfaces in $\mathbb{H}^{n+1}$, with structural gradient and curvature bounds derived directly from the support property, obviating the standard sign assumptions required in Weingarten-type curvature problems (Chen et al., 2020).

• Quermassintegral Inequalities (Alexandrov–Fenchel Family):

On the sphere, horo-convex domains satisfy the full suite of quermassintegral inequalities. The classical Guan/Li inverse curvature flow preserves horo-convexity, and the long-time behavior establishes monotonicity and rigidity in the inequalities, parallel to the hyperbolic and Euclidean settings (Pan et al., 14 Dec 2025).

• Metric Comparison:

In the Poincaré disk, horo-convex domains admit sharp lower bounds for the hyperbolic metric in terms of distance to the boundary horocycle. This provides geometric control and identifies extremal domains (horocrescents) for which equality is attained (Arango et al., 31 Jul 2024).

5. Internal, External, and Projective Criteria

Several intrinsic and extrinsic characterizations elucidate the nature of horo-convexity:

• Planar Model Internal Characterization:

Within the unit disk, a domain is horo-convex if and only if every pair of points can be joined by a finite sequence of $C^1$ arcs, each with hyperbolic curvature in $(-2,2)$, i.e., by admissible Jordan arcs generalizing geodesic segments. This links the external support property (by horodisks) to an internal geometric chain condition (Arango et al., 31 Jul 2024).

• Comparison Principle and Avoidance:

For horo-convex hypersurfaces evolving under inverse curvature flows, a maximum principle ensures that two such hypersurfaces can touch only if they coincide entirely (the avoidance principle). This principle underpins rigidity, uniqueness, and stability results (Pan et al., 14 Dec 2025).

6. Examples and Non-Examples

The following table summarizes canonical examples and distinguishes horo-convexity from weaker forms:

Setting	Horo-Convex Examples	Non-Examples
$\mathbb{H}^n$	Geodesic balls, horoballs, finite intersections	Geodesically convex "nosed" domains with some κ<1
$S^{n+1}$	Spheres tangent to equator (horo-spheres); domains lying locally inside such spheres	Convex surfaces touching or crossing equator
Hadamard manifold functions	Scaled Busemann functions, Fréchet mean, minimal enclosing ball	Functions lacking Busemann support inequality
$\mathbb{D}$	Horo-crescents (complements of horodisks)	Domains not admitting internal chain condition

• For hypersurfaces in $\mathbb{H}^{n+1}$: horospheres are the borderline case ($\kappa_i \equiv 1$); geodesic spheres with $r>0$ are strictly horo-convex ($\kappa_i > 1$); outward-normal perturbations reducing curvature at any point immediately destroy horo-convexity, even if geodesic convexity is preserved (Wei et al., 15 Oct 2025, Chen et al., 2020).

7. Research Impact, Open Problems, and Future Directions

Horo-convexity has crystallized as a vital structure underpinning modern analysis, geometry, and optimization in negatively curved and projective settings:

• It restores curvature-independent rates for first-order optimization on Hadamard manifolds, challenging prior bottlenecks observed for g-convex minimization (Criscitiello et al., 22 May 2025).
• It underlies sharp functional inequalities, spectrally optimal domains, and metric geometry comparisons, often with rigidity (uniqueness) tightly characterized via horo-convexity conditions (Wei et al., 15 Oct 2025, Pan et al., 14 Dec 2025).
• Open directions include determining explicit diameter thresholds for super log-concavity, extension to broader classes of partial differential equations, analogues for Brascamp–Lieb-type inequalities, lengthening the allowable class of boundary regularity (beyond $C^2$), and analyzing the stability of concavity under small geometric perturbations (Wei et al., 15 Oct 2025).

Horo-convexity thereby provides both a geometric and analytic bridge from Euclidean to singular and non-Euclidean ambient spaces, with techniques and consequences permeating theory and computation across geometry, analysis, and optimization.