On the smoothness of the equizonal ovaloids
The generalized equal area zones property demands that the \(n\)-volume of a ``hyperzone'' sliced out of the hypersurface of revolution by two parallel hyperplanes orthogonal to the axis of rotation be proportional to the distance between the hyperplanes. For each \(n \geq 2\), there is, up to scaling, a unique closed strictly convex \(n\)-dimensional hypersurface of revolution in \(\mathbb R^{n+1},\) of class \(C^1\), satisfying the generalized equal area zones property. When \(n=2\), this object is the ordinary sphere, but for \(n>2\) it is a more exotic object, the equizonal \(n\)-ovaloid \(\mathrm{EO}^n\). Dodd and Coll in 2008 reported that if \(n\) is even, then \(\mathrm{EO}^n\) is of class \(C^\infty\), and if \(n\) is odd, \(\mathrm{EO}^n\) is of class \(C^{(n-1)/2}\). The authors show that if \(n\) is even, \(\mathrm{EO}^n\) is analytic, and that if \(n\) is odd, \(\mathrm{EO}^n\) is analytic everywhere except at its two poles where it is of class exactly \(C^{n-1}\).
scientific article; zbMATH DE number 6151651