Let *ω* be a unit tangent vector to a smooth complete simply connected manifold without conjugate points \(W\in \mathcal {W}^{n}\) at a point *p*∈*W*. Let *α* be the unique geodesic with tangent *ω*=*α*^{′}(0) and *p*=*α*(0). The Busemann function \(b_{\omega }:W\in \mathcal {W}^{n}\rightarrow \mathbb {R} \) is defined by

$$ b_{\omega }\left(x\right) ={\lim}_{t\rightarrow \infty }\left[ t-d\left(x,\alpha \left(t\right) \right) \right], $$

(1)

where *d* is the distance function. The right hand side is well-defined and the Busemann function *b*_{ω} is smooth in a complete simply connected manifold without conjugate point \(W\in \mathcal {W}^{n}\) whereas *b*_{ω} is at least *C*^{2} given that *W* has no focal points(see [1, Theorem 2]). The level set of a Busemann function, that is \(b_{\omega }^{-1}\left (0\right) \), is called a horosphere *H*_{ω}(*p*) where *p*=*α*(0). Likewise, the open and the closed horoballs in \(W\in \mathcal {W}^{n}\) are defined as the sets \(D_{\omega }\left (p\right) =b_{\omega }^{-1}\left (\left (0,\infty \right) \right) \) and \(\bar {D}_{\omega }\left (p\right) =b_{\omega }^{-1}\left (\left [ 0,\infty \right) \right) \) respectively. Let *α*(*t*) be the geodesic passing through a point \(p\in W\in \mathcal {W}^{n}\) with *α*^{′}(0)=*ω*. It is well-known that the horosphere *H*_{ω}(*p*) is the limit of the geodesic spheres *S*(*α*(*t*),*t*) passing through *p*=*α*(0) and having center *α*(*t*) as *t*→*∞*. The horospheres *H*_{u}(*p*),*u*=*α*^{′}(0),*p*=*α*(0) and *H*_{v}(*q*)*v*=*α*^{′}(*a*),*q*=*α*(*a*) are called co-directional or parallel horospheres and parallel horospheres touch each other at infinity. Notice that the horopsheres *H*_{ω}(*p*) and *H*_{−ω}(*p*) have *p* as their unique common point; otherwise, they coincide. Hyperplanes are horospheres in the Euclidean space *E*^{n}. The horosphere *H*_{ω}(*p*) with a given direction *ω* and a given point *p* is unique. Finally, the horospheres, as a level surfaces of a Busemann function, are equidistant family of surfaces whose orthogonal trajectories are geodesics.

It is noted that, thanks to the well-known Hopf-Rinow theorem, there is a length minimizing geodesic segment joining each pair of points in a complete connected Riemannian manifold *W*. If, in addition, *W* is simply connected and has no conjugate points, then the exponential map is a covering map and each pair of points is joined by a unique and hence minimal geodesic(see Section 10.7 of [2]). Finally, a set *A* of \(W\in \mathcal {W} ^{n}\) is compact if and only if it is closed and bounded. All manifolds with negative curvature are members of \(\mathcal {W}^{n}\). For example, the hyperbolic Poincare upper half-plane model

$$H^{2}=\{(x,y)\in \mathbb{R}^{2}:y>0\}, $$

equipped with the metric *g*_{11}=*g*_{22}=*y*^{−2} and *g*_{12}=0 lies in \(\mathcal {W}^{2}\) (see [3] for more details); however, the unit sphere *S*^{2} does not lie in \(\mathcal {W}^{2}\) since all antipodal points are conjugate points.

A subset *A* of \(W\in \mathcal {W}^{n}\) is convex if the geodesic segment [*p**q*] joining any two points *p*,*q*∈*A* lies in *A*. Three different definitions of convex sets in general Riemannian manifolds were studied in [4]. The whole manifold *W* geodesics are all convex sets. Also, open and closed geodesic balls of manifolds with negative curvature are convex sets. On the other hand, the union of two different geodesics is not convex and the complement of a convex set is not necessarily convex. Note that the existence and uniqueness of geodesic segments in these manifolds is trivial; however, for example, the whole sphere *S*^{n} is not convex since antipodal points have many minimal geodesic segments joining them. Convex functions are also deeply studied in Riemannian geometry (the reader is referred to [5] for a detailed study of convex functions on manifolds with negative curvature).

Let *p* be a point in a complete simply connected manifold without conjugate point \( W\in \mathcal {W}^{n}\). The point *p* has a foot point *f* in subset *A* of *W* if the distance function \(l:A\rightarrow \mathbb {R}\) defined by *l*(*x*)=*d*(*p*,*x*),*x*∈*A* attains its minimum at *f*. The point *p* is said to have a farthest point *F* in *A* if the function *l* attains its maximum at *F* [6, 7]. The geodesic ray starting at *p* and passing through *q* is denoted by *R*(*p**q*), and the entire geodesic passing through them is denoted by *G*(*p**q*).

Convex sets, foot, and farthest points play a very important role in both convex analysis and optimization (see for example [8–10] and references therein). Generalizations and extensions of convex sets and their separation and supporting surfaces are of particular interest [11, 12]. Each pair of points in a simply connected smooth Riemannian manifold without conjugate points has a unique and hence minimal geodesic joining them whereas manifolds without focal points has convex geodesic spheres [13–17]. It is well-known that the class of complete simply connected manifolds without focal points is a proper subclass of \(\mathcal {W}^{n}\). Manifolds with non-positive sectional curvatures have no focal points [18–22]. Horospheres and totally geodesic hypersurfaces in \(W\in \mathcal {W}^{n}\) play a significant role in defining both supporting and separation theorems for convex sets.

In this note, the concepts of separation and horosphere slab separation of convex sets are studied in \(W\in \mathcal {W}^{n}\). Sufficient conditions for two disjoint closed convex sets to be separated by a slab of horosheres are given. Foot and farthest points of a convex set in \(W\in \mathcal {W}^{n}\) are considered.