Equidistribution rate for Fekete points on some real manifolds

Let $L$ be a positive line bundle over a compact complex projective manifold $X$ and $K\subset X$ be a compact set 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 $k\rightarrow\infty$. 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 ${\cal C}^5$-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 ${\Bbb R}^n$ or the unit sphere in ${\Bbb R}^{n+1}$.