Positive centers and the Bonnesen inequality

Abstract:

A positive center of a convex curve is a point from which the function $p(\theta )L - A - \pi p{(\theta )^2}$ is positive for all values of $\theta$. The support function is $p$ and $L$ and $A$ 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.