Rotational Sweeps of Compact Connected Sets: A Geometric Route to Enclosing Annuli

Given a set K in Euclidean space and a point p, consider the union of all copies of K obtained by rotating K through every possible orientation about p; this note calls that union the rotational sweep of K about p. The sweep is determined entirely by the set of distances from p to K. When K is compact and connected, those distances form an interval, so the sweep is exactly a circular annulus in the plane and a spherical shell in higher dimensions; in the planar case its area is π times the difference between the squared outer and inner radii. The associated squared annular-span functional is convex in every Euclidean dimension. For a finite planar point sample, the analogous minimum-area enclosing-annulus problem has a four-variable linear-programming formulation already present in the computational-geometry literature. When the pivot is constrained to a compact convex set, minimizing planar swept area reduces to the classical minimum enclosing disk problem. The purpose of the note is expository and structural. It does not claim that the minimum-area enclosing-annulus problem is new.