Constant width and transnormality in spheres (Q1919300)

Let \(C\) be a body of constant width \((\pi/2) + x\in ((\pi/2),\pi)\) in the unit sphere \(S^n\). Then \(C\) contains a body \(C^*\) of constant width \((\pi/2)-x\) such that \(C\) is the closed \(x\)-neighborhood of \(C^*\). Moreover, the body \(C\) is smooth. Throughout the text some properties on convex sets of constant width on the \(n\)-sphere are given.