Dark clouds on spheres and totally non-spherical bodies of constant breadth (Q1403521)

Continuing investigations of H. G. Eggleston and B. Weissbach, the authors prove the following: For any \(d\geq 3\), there exists a \(d\)-dimensional body of constant width in Euclidean \(d\)-space such that any of its two-dimensional projections is not spherical. Calling such a body of constant width totally non-spherical, they show that the circumradius of each two-dimensional projection of any totally non-spherical body \(C\) of constant width is larger than half the diameter of \(C\). As a tool they use the concept of dark clouds.