Symmetrical Types of Convex Regions

Abstract

A region C in the plane, or in vector space of any number of dimensions, is convex f whenever P, Q are points of C, the line segment PQ is entirely contained in C. A region S is starlike if there is a point 0 of S, such that for every direction, the intersection with S of the ray or half-line from 0 in the direction is either the entire ray, a segment with 0 as one end-point, or only the point 0 (as may be the case if 0 is on the boundary of S). To facilitate analytic rather than synthetic study of symmetry, in this paper socalled gauge-functions areassociated with convex and starlike regions. For examples, the interior of a triangle is an open convex region; the interior plus the edges and vertices is a closed convex region. The interior of any non-convex quadrilateral, not having intersecting sides, is a starlike region. The gauge-function for an ellipse C is p(x) = [(xl/al)2 + (x2/a2 )2]i; C is the set of all points x for which p(x) < 1. A plane region R has irnvolutory symmetry in the origin if for every point x = (x1,x2) of R, the point (-x1, -x 2) is also a point of R. Unlike a circle, ellipse, or parallelogram, a triangle does not possess involutory symmetry in a point. An ellipsoid with two equal semi-axes is generated by rotation of an ellipse about the third axis, and therefore hlas complete rotational symmetry about the axis. In this paper, we explore the possible types and interrelations of rotational and involutory symmetries which may be possessed by regions in n dimensional space. Plane and solid analytic geometry, elementary group theory, and some acquaintance with matrix algebra, or study of reference [2], are suffic-ient for understanding most of the paper. For full appreciation of section 2, however, it would be desirable to have a knowledge of thie elements of the theory of normed vector soaces.