isodiametric

Isodiametric refers to a relation between the diameter of a set and its measure (area in the plane, volume in higher dimensions). The term is most often used in connection with the isodiametric inequality, which states that among all plane figures with a given diameter, the circle has the greatest area.

Formally, for any planar set A with diameter D, its area satisfies area(A) ≤ π D^2 / 4, with equality precisely when A is a disk of diameter D. A set that attains this bound is called isodiametric. The concept generalizes: in n-dimensional Euclidean space, among all bodies with diameter D, the volume is at most κ_n (D/2)^n, where κ_n is the volume of the unit ball in R^n; equality holds exactly for n-balls.

The isodiametric constant Id_n is the supremum of vol(A)/diam(A)^n over all sets A with positive measure; in Euclidean n-space, Id_n = vol(B^n)/2^n. The inequality and its equality case were established in the plane by Blaschke and Bieberbach in the early 20th century and are foundational in geometric measure theory and convex geometry.

In metric geometry, the isodiametric problem asks, in a given metric space, what is the maximum possible measure of a subset with a given diameter; the answer depends on the space and the measure. The term isodiametric thus describes both the extremal shapes (isodiametric sets) and the general inequality relating diameter and measure.