Applications of spherical designs to Banach space theory
Spherical designs constitute sets of points distributed on spheres in a regular way. They can be used to construct finite-dimensional normed spaces which are extreme in some sense: having large projection constants, big or small Banach-Mazur distance to Hilbert spaces or $\ell_p$-spaces. These examples provide concrete illustrations of results obtained by more powerful probabilistic techniques which, however, do not exhibit explicit examples. We give a survey of such constructions where the geometric invariants can be estimated quite precisely.