Let X be a compact K\"ahler manifold with a given ample line bundle L. In
\cite{Don05}, Donaldson proved that the Calabi energy of a K\"ahler metric in
c1β(L) is bounded from below by the supremum of a normalized version of the
minus Donaldson--Futaki invariants of test configurations of (X,L). He also
conjectured that the bound is sharp. In this paper, we prove a metric analogue
of Donaldson's conjecture, we show that if we enlarge the space of test
configurations to the space of geodesic rays in E2 and replace the
Donaldson--Futaki invariant by the radial Mabuchi K-energy M, then a
similar bound holds and the bound is indeed sharp. Moreover, we construct
explicitly a minimizer of M. On a Fano manifold, a similar sharp
bound for the Ricci--Calabi energy is also derived.Comment: Final version. Statement of Theorem 4.1 corrected. To appear on
Analysis & PD