For fixed w in (0, pi) excluding pi/2, the liminf and limsup of the infimal relative effective radius of constant-width bodies on S^n are bounded between positive explicit constants sigma_l(w) and sigma_u(w) both strictly less than 1.
Schramm,On the volume of sets having constant width, Israel Journal of Mathematics63(1988), no
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On the spherical Blaschke-Lebesgue problem
For fixed w in (0, pi) excluding pi/2, the liminf and limsup of the infimal relative effective radius of constant-width bodies on S^n are bounded between positive explicit constants sigma_l(w) and sigma_u(w) both strictly less than 1.