Maße für zyklische symmetrien ebener konvexer körper

Abstract

Let

denote a finite group of rotations of

A realvalued functional f defined on a subfamily

of the family of all convex bodies in

is called a measure of symmetry with respect to

if (1) 0⩽(K)⩽1 for all

; (2) f(K)=1 iff K is symmetric with respect to

; (3)

for all

and for all orientation preserving similarties τ of

; (4) f:

is continuous.

A special measure of symmetry with respect to

,

, is introduced. The following theorems hold: Let

denote the cyclic group of proper rotations of

generated by the rotation about 2π/k with the origin as center, k⩾3 an integer, and let

denote the family of all plane centrally symmetric convex bodies containing at least two points.

• 1.

  

  for k⩾3, where c_k is a positive real number which is explicitly calculated.

• 2.

  

  iff K is a segment. For K=3,4 better results can be obtained. One tool for these improvements is:

• 3. About every plane convex body with interior points an equilateral triangle can be circumscribed such that its center of gravity lies in the interior of K.