Rényi Entropies of Projections

In this paper we are interested in n-dimensional uniform distributions on a triangle and a sphere. We show that their marginal distributions are maximizers of Renyi entropy under a constraint of variance and expectation in the respective cases of the sphere and of the triangle. Moreover, using an example, we show that a distribution on a triangle with (uniform) maximum entropy marginals may have an arbitrary small entropy. As a last result, we address the asymptotic behavior of these results and provide a link to the de Finetti theorem