Approximation of spherical convex bodies of constant width π/2

Let C⊂ S² be a spherical convex body of constant width τ. It is known that (i) if τ<π/2 then for any ε>0 there exists a spherical convex body C_ε of constant width τ whose boundary consists only of arcs of circles of radius τ such that the Hausdorff distance between C and C_ε is at most ε; (ii) if τ>π/2 then for any ε>0 there exists a spherical convex body C_ε of constant width τ whose boundary consists only of arcs of circles of radius τ-π/2 and great circle arcs such that the Hausdorff distance between C and C_ε is at most ε. In this paper, we present an approximation of the remaining case τ=π/2, that is, if τ=π/2 then for any ε>0 there exists a spherical polytope P_ε of constant width π/2 such that the Hausdorff distance between C and P_ε is at most ε.