---
title: Horospherical Convexity (h-convexity) in Hyperbolic Spaces
url: https://www.emergentmind.com/topics/horospherical-convexity-h-convexity
type: topic
---

# Horospherical Convexity (h-convexity) in Hyperbolic Spaces

Horospherical convexity, usually abbreviated **h-convexity**, is a hyperbolic analogue of Euclidean convexity in which affine half-spaces are replaced by horoballs, horospheres, or Busemann-function sublevel sets. In hyperbolic space, a bounded domain \(K\subset \mathbb H^{n+1}\) is h-convex if every boundary point admits a supporting horoball; for smooth boundaries this is equivalent to the principal-curvature condition \(\kappa_i\ge 1\), while strict or uniform h-convexity requires \(\kappa_i>1\) [2411.17328]. Closely related variants appear for immersed hypersurfaces, planar horocyclically convex domains, and Hadamard-manifold optimization, all built from the same principle: the supporting objects are adapted to the asymptotic geometry of nonpositively curved spaces rather than to linear structure [1611.06421][2407.21271][2505.16970].

## 1. Supporting horoballs and curvature thresholds

The basic geometric definition in \(\mathbb H^{n+1}\) is a support condition. A bounded domain \(K\subset \mathbb H^{n+1}\) is h-convex if every boundary point of \(\partial K\) admits a supporting horoball, that is, a horoball \(B\) with \(K\subset B\) and the boundary point lying on \(\partial B\). Geometrically, this means that the domain is an intersection of horoballs, just as a Euclidean convex body is an intersection of half-spaces. In the hyperboloid model, horospheres are the hypersurfaces whose principal curvatures are all equal to \(1\), so for a smooth boundary the support condition is equivalent to
\[
\kappa_i\ge 1,\qquad i=1,\dots,n,
\]
and strict h-convexity to
\[
\kappa_i>1,\qquad i=1,\dots,n.
\]
This curvature threshold is the hyperbolic shift that distinguishes horospherical convexity from ordinary geodesic convexity [2411.17328][1708.09583].

A weaker immersion-theoretic version is **weak horospherical convexity**. For an oriented immersed hypersurface \(\phi:M^n\to\mathbb H^{n+1}\), one considers the tangent horosphere \(\mathcal H_p\) at \(\phi(p)\) whose outward unit normal agrees with the chosen normal field. The hypersurface is weakly horospherically convex at \(p\) if a neighborhood of \(p\) stays on one side of \(\mathcal H_p\), equivalently if all principal curvatures satisfy either
\[
\kappa_i(p)>-1\quad\text{for all }i
\qquad\text{or}\qquad
\kappa_i(p)<-1\quad\text{for all }i.
\]
The corresponding uniform version requires a constant \(\kappa_0>-1\) with \(\kappa_i\ge \kappa_0\) everywhere, and this is the class used in the global correspondence and embeddedness theory [1611.06421].

This support viewpoint also has lower-dimensional analogues. In the unit disk \(\mathbb D\), a proper subdomain \(G\subsetneq\mathbb D\) is **horocyclically convex** if every interior boundary point admits a supporting horodisk. The paper on horocyclically convex domains emphasizes the analogy
\[
\text{Euclidean convexity} \leftrightarrow \text{supporting half-planes},\qquad
\text{hyperbolic convexity} \leftrightarrow \text{supporting hyperbolic half-planes},\qquad
\text{horo-convexity} \leftrightarrow \text{supporting horodisks},
\]
and states that every \(h\)-convex domain is horo-convex [2407.21271].

## 2. Support functions, Gauss maps, and conformal correspondences

For smooth uniformly h-convex hypersurfaces, horospherical convexity is encoded by a support function on the sphere. If \(X\) is the position vector of \(M=\partial K\subset \mathbb H^{n+1}\) and \(v\) is the outward unit normal, then
\[
X-v=e^{-u}(x,1)
\]
for some \(u\in\mathbb R\) and \(x\in\mathbb S^n\). The function
\[
\mathbf y=e^u
\]
is the **horospherical support function**, and the associated horospherical Gauss map \(G:\partial K\to\mathbb S^n\) sends each boundary point to the direction \(x\). For uniformly h-convex domains this Gauss map is a diffeomorphism, so the hypersurface can be reparametrized over \(\mathbb S^n\). In this parametrization the decisive tensor is
\[
A_{ij}[\mathbf y]
=
y_{ij}
-\frac{|\nabla y|^2}{2y}\,g_{ij}
+\frac12\Bigl(y-\frac1y\Bigr)g_{ij},
\]
and uniform h-convexity is equivalent to
\[
A[\mathbf y]>0.
\]
Moreover,
\[
(h_i^{\ j}-\delta_i^{\ j})\,A_{jk}[\mathbf y]=\delta_{ik},
\]
so the eigenvalues of \(A[\mathbf y]\) are the hyperbolic curvature radii
\[
R_i=\frac{1}{\kappa_i-1}.
\]
The horospherical surface area measure is then
\[
dS(K,x)=\det A[\mathbf y(x)]\,d\sigma_{\mathbb S^n},
\]
and the \(k\)-th horospherical \(p\)-surface area measure is
\[
dS_{p,k}(K,x)=C_{n,k}\,y(x)^{-p-k}\,\sigma_{n-k}\!\bigl(A[\mathbf y(x)]\bigr)\,d\sigma_{\mathbb S^n}.
\]
These formulas make h-convexity directly usable in fully nonlinear PDE [2411.17328].

A parallel formulation exists for weakly horospherically convex hypersurfaces via the light-cone map. For an oriented immersion \(\phi:M^n\to\mathbb H^{n+1}\) with unit normal \(\eta\), the light-cone map is
\[
\psi=\phi-\eta:M\to\mathbb N^{n+1}_+.
\]
If \(G\) is the hyperbolic Gauss map, then
\[
\psi=e^{\tilde\rho}(1,G),
\]
where \(\tilde\rho\) is the horospherical support function, and the induced **horospherical metric** is
\[
g_h=\psi^*\langle,\rangle=e^{2\tilde\rho}\,G^*g_{\mathbb S^n}.
\]
When \(G\) is injective, this becomes a conformal metric \(\hat g=e^{2\rho}g_{\mathbb S^n}\) on a domain \(\Omega\subset\mathbb S^n\). The local correspondence is governed by the tensor
\[
P=-\nabla^2\rho+d\rho\otimes d\rho-\frac12\bigl(|\nabla\rho|^2-1\bigr)g_{\mathbb S^n},
\]
whose eigenvalues \

Source: https://www.emergentmind.com/topics/horospherical-convexity-h-convexity