  The long-standing problem of minimal projections is addressed from a
computational point of view. Techniques to determine bounds on the projection
constants of univariate polynomial spaces are presented. The upper bound,
produced by a linear program, and the lower bound, produced by a semidefinite
program exploiting the method of moments, are often close enough to deduce the
projection constant with reasonable accuracy. The implementation of these
programs makes it possible to find the projection constant of several
three-dimensional spaces with five digits of accuracy, as well as the
projection constants of the spaces of cubic, quartic, and quintic polynomials
with four digits of accuracy. Beliefs about uniqueness and shape-preservation
of minimal projections are contested along the way.
