  Let L be a positive line bundle over a compact complex projective manifold X
and K be a compact subset of X which is regular in a sense of pluripotential
theory. A Fekete configuration of order k is a finite subset of K maximizing a
Vandermonde type determinant associated with the power L^k of L. Berman,
Boucksom and Witt Nystr\"om proved that the empirical measure associated with a
Fekete configuration converges to the equilibrium measure of K as k tends to
infinity. Dinh, Ma and Nguyen obtained an estimate for the rate of convergence.
Using techniques from Cauchy-Riemann geometry, we show that the last result
holds when K is a real nondegenerate C^5-piecewise submanifold of X such that
its tangent space at any regular point is not contained in a complex hyperplane
of the tangent space of X at that point. In particular, the estimate holds for
Fekete points on some compact sets in R^n or the unit sphere in R^{n+1}.
