On the Axial Symmetry of 2D Star-Shaped Sets

An essential aspect of the study of shapes is the symmetry because of its importance from a theoretical point of view and its applicability in multiple real-life problems. In this manuscript, the axial symmetry of 2D star-shaped sets is analyzed. For such a purpose, different measures of axial symmetry of a star-shaped set are proposed and the concept of a best symmetry axis is also introduced. By means of them, families of symmetry measures for star-shaped sets quantifying the degree of symmetry of a set of that class are introduced. All of them are discussed in detail, providing their main properties and the existence of at least a best axis of symmetry, which could be not unique, for any star-shaped set. Some examples illustrate the concepts and results of the manuscript.