A Bonnesen-style inradius inequality in 3-space (Q1111158)

Let K be a convex body in 3-dimensional Euclidean space, and let V, S, (1/2\(\pi)\)M, and r denote the volume, surface area, mean-width and inradius of K. The author establishes the inequality \[ V - rS + \frac{2}{3}r^ 2M \leq 0 \] with equality if and only if K is a cap body of a ball. A cap body of a ball is the convex hull of the ball and countably many points exterior to it such that the line segment joining any two of these points intersects the ball. The above inequality is obtained as a consequence of an inequality providing a lower bound for the volume of an inner parallel body of K.