  We investigate projection constants within classes of multivariate
polynomials over finite-dimensional real Hilbert spaces. Specifically, we
consider the projection constant for spaces of spherical harmonics and spaces
of homogeneous polynomials as well as for spaces of polynomials of finite
degree on the unit sphere. We establish a connection between these quantities
and certain weighted $L_1$-norms of specific Jacobi polynomials. As a
consequence, we present exact formulas, computable expressions and
asymptotically accurate estimates for them. The real case we address is
considerably more nuanced than its complex counterpart.
