Horosphere slab separation theorems in manifolds without conjugate points

Abstract

Abstract Let Wn $\mathcal {W}^{n}$ be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let W∈Wn $W\in \mathcal {W}^{n}$ and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in W∈W $W\in \mathcal {W}$ are considered.

Keywords

• Horosphere separation
• Slab horosphere separation
• Manifolds without conjugate points
• Separation of convex sets