Properties of External Visibility. 1
The external visibility of a closed set S means the visibility referred to its complementary points. This kind of geometrical study appears naturally in the planning of movements of servomechanisms and robots. The aim of this paper is to connect the external visibility of a certain set S (in particular Stavrakas' half-line property) with new properties which involve points of S instead of points of its complement. We say that a hunk S has the shining boundary property if its complement is free from bounded connected components and for each boundary point of S there exists a ray issuing from it and disjoint with the interior of S. It is proved here the equivalence (for a planar hunk) of this property and Stavrakas' half-line property. Furthermore, in some cases which we specify, Stavrakas' property is equivalent to the fact that each boundary point has nontrivial strong inner stem. These equivalencies yield new versions of some characterizations of starshapedness due to Stavrakas. § 1.BASIC DEiINITIONS AND NOTATIONS Unless otherwise stated, all the points and sets considered here are included in Rb the real ndimensional euclidean space. The interior, closure, boundary, and complement of a set S are denoted by: intS, cl S, bdry S, CS respectively. The open segment joining x and y is denoted (x,y). The substitution of one or both parentheses by square ones indicates the adjunction of the corresponding extremes. We say that x sees y via S if [x,y] c S. The star of x in S is the 1 The preparation of this paper was supported in part by Comisi6n de Investigaciones Cientificas de la Pcia. de Bs. AS.
Critical visibility and outward rays
The notion ofclear visibility has appeared recently in the literature concerning the geometry of starshaped sets. A somewhat converse notion, that ofcritical visibility, is explored here. As a by-product it is proved that the points of local nonconvexity of a closed connected set S determine the nontrivial portions of the boundary of the star of a point in S. Paragraph 4 is devoted to the study of the concepts ofoutward ray through a boundary point andinner stem of that point. A new characterization of the convex kernel of a starshaped set results from this study, and Krasnoselsky-type theorems are obtained in the planar case.
A CLASS OF ALMOST CONTACT RIEMANNIAN MANIFOLDS
Recently S. Tanno has classified connected almostcontact Riemannian manifolds whose automorphism groups have themaximum dimension [9]. In his classification table the almost contactRiemannian manifolds are divided into three classes: (1) homogeneousnormal contact Riemannian manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain