On curves with circles as their isoptics

In the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.

On the Geodesic Identification of Vertices in Convex Plane Graphs

A shortest path between two vertices u and v in a connected graph G is a u − v geodesic. A vertex w of G performs the geodesic identification for the vertices in a pair u , v if either v belongs to a u − w geodesic or u belongs to a v − w geodesic. The minimum number of vertices performing the geodesic identification for each pair of vertices in G is called the strong metric dimension of G . In this paper, we solve the strong metric dimension problem for three convex plane graphs by performing the geodesic identification of their vertices.

Some Inequalities for Convex Sets

The paper concerns inequalities between fundamental quantities as area, perimeter, diameter and width for convex plane fugures.