A four‐dimensional body of constant width

Abstract

The study of bodies of constant width is a classical subject in convex geometry, with the three‐dimensional Meissner bodies being canonical examples. This paper presents a novel geometric construction of a body of constant width in , addressing the challenge of constructing such bodies in higher dimensions. Our method produces a natural analogue of the second Meissner body by modifying a four‐dimensional Reuleaux simplex. The resulting body possesses tetrahedral symmetry and has a boundary composed of both smooth surfaces and a nonsmooth subset of the Reuleaux 4‐simplex. Furthermore, we analyze the orthogonal projection of this body onto the three‐dimensional hyperplane of its base. This “shadow” is a three‐dimensional body of constant width with tetrahedral symmetry. It has six elliptical edges and its volume is only slightly larger than that of the Meissner bodies. This body was recently constructed as a projection of a different four‐dimensional body, however the construction presented here is new and gives additional properties.