Convexidad en el plano hiperbólico

Santaló, Lluís
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A set of points Q in the hyperbolic plane is said to be h-convex (convex with respect to horocycles) if for each pair of points A, B belonging to Q, the entire segments of the two horocycles determined by A, B belong to Q. In this paper we give the following properties for h-convex sets: 1. If Q is an h- convex, the whole horocyclic lentil bounded by the horocycles O, O' determined by two points A, B of Q belongs to Q and we get the inequalities (2 . 8) between the length L of the boundary Q, the area F, the width A and the diameter D of Q. 2. If kg. denotes the geodesic curvature of the boundary Q of a set Q, then Q is an h-convex if and only if xg > 1. 3. For each point A of the boundary ‘Q of an h-convex set Q we consider the two tangent horocycles O- and O+ and define the breadth of Q with respect to each of them. These h-breadths O-, n+ satisfies the formulae (5.3) and (5.4). 4. The sets of constant breadth are considered specially. For them the formula (6.1) holds the last theorem states that the Reuleaux triangle has minimal length and minimal area among all the sets of the same constant breadth of the hyperbolic plane ​
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