  For closed connected Riemannian spin manifolds an upper estimate of the
smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius
is proved. When combined with known lower Dirac eigenvalue estimates, this has
a number of geometric consequences. Some are known and include Llarull's scalar
curvature rigidity of the standard metric on the sphere, Geroch's conjecture on
the impossibility of positive scalar curvature on tori and a mean curvature
estimate for spin fill-ins with nonnegative scalar curvature due to Gromov,
including its rigidity statement recently proved by Cecchini, Hirsch and
Zeidler. New applications provide a comparison of the hyperspherical radius
with the Yamabe constant and improved estimates of the hyperspherical radius
for K\"ahler manifolds, K\"ahler-Einstein manifolds, quaternionic K\"ahler
manifolds and manifolds with a harmonic 1-form of constant length.
