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Notions of distances between metric (measure) spaces such as the Gromov-Hausdorff distance and its Optimal Transport variants are nowadays often invoked in applications related to data classification. Interestingly, the precise value of these distances on pairs of canonical shapes is known only in very limited cases. In this talk, I will describe lower bounds for the Gromov-Hausdorff distance between spheres (endowed with their geodesic distances) which we prove to be tight in some cases via the construction of optimal correspondences.  These lower bounds arise from applying a certain version of the Borsuk-Ulam theorem for discontinuous functions.