Invent. math. 69, 253-257 (1982) Illvefltlolles mathematicae (() Springer-Verlag 1982 Many Endpoints and Few Interior Points of Geodesics* Tudor Zamfirescu Abteilung Mathematik, Universifiit Dortmund, Posffach 500500, 4600 Dortmund, Bundesrepublik Deutschland This paper is about abnormal convex surfaces and their equally abnormal geodesics. We do not feel ashamed of studying them because, in the sense of Baire categories, most convex surfaces are abnormal! Thus, in fact, they should be considered normal and vice-versa. By a convex surface we always mean a closed one (see Busemann [2], p. 3), by a segment a shortest path on the surface ([2], p. 75), by a geodesic a curve which is locally a segment (see for a precise definition [2], p. 77). In spaces of second Baire category, we use the words most and typical in the sense of "all, except those in a set of first Baire category". The space of all convex surfaces in IR", endowed with Hausdorfl's metric, is a Baire space. We shall see how abnormal convex surfaces may be, by proving that most of them are so. Results Since any two points of a convex surface are joined by at least one segment, the union of all segments equals the surface. A point of a segment different from its two endpoints will briefly be called interior. Is each point of a surface an interior point'? The answer is easy for non-smooth surfaces: no conical point is (for any segment) an interior point ([1], p. 155). Points which are not for any segment interior will be called endpoints. They are, of course, endpoints of lots of geodesics. Smooth surfaces with an endpoint are also known ([1], p. 58- 59). But, for each convex surface of class C 2, every point is an interior point of a segment in each tangent direction. More generally, this happens at a point x if the lower indicatrix at every point y in some neighbourhood of x does not contain y as a boundary point (Busemann [2], p. 92). Clearly, the set of all interior points is uncountable and dense, for an arbitrary convex surface. Thus, it seems that, usually, convex surfaces must have many interior points. But let us look more closely at a typical convex surface: it is of class C 1, but * Dedicated to Th. Hangan 0020-9910/82/0069/0253/$01.00