Positive centers and the Bonnesen inequality

Abstract

A positive center of a convex curve is a point from which the function p ( θ ) L − A − π p ( θ ) 2 p(\theta )L - A - \pi p{(\theta )^2} is positive for all values of θ \theta . The support function is p p and L L and A A are the length and area of the curve, respectively. This paper proves that all convex curves have a positive center and gives an example which shows that the common centroids (Steiner point, etc.) are not necessarily positive centers. A strengthened version of the Bonnesen inequality is obtained and a simplified proof of the one-dimensional Firey inequality.