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It has been recently shown that a damped Newton algorithm allows to solve the optimal transport problem between an absolutely continuous measure and a discrete one when the cost satisfies a discrete version of the Ma-Trudinger-Wang condition. I will consider here the case where the source measure is not supported on a set of maximal dimension. More precisely, under genericity and connectedness conditions, I will show the convergence of the damped Newton algorithm to solve the optimal transport problem for the quadratic cost in $\mathbb{R}^d$ when the source measure is supported on a simplex soup, each simplex being of arbitrary dimension greater than $2$.