REDUCED CONVEX-BODIES IN THE PLANE

A convex body R of Euclidean d-space E d is called reduced if there is no convex body properly contained in R of thickness equal to the thickness Δ(R) of R. The paper presents basic properties of reduced bodies in E 2. Particularly, it is shown that the diameter of a reduced body R⊂E 2 is not greater than √2Δ(R), and that the perimeter is at most (2+1/2π)Δ(R). Both the estimates are the best possible. © 1990 The Weizmann Science Press of Israel.