Consider a vector with n independent random coordinates uniformly distributed in [-1/2,1/2] (so that the density is 1). It is known that the density of any k-dimensional marginal of this vector is uniformly bounded above by a function depending only on k. A uniform lower bound on the marginal density is impossible. Indeed, the marginal density at a point having distance greater than 1/2 from the origin can be zero. We show that if this distance does not exceed 1/2, then the marginal density is lower bounded by a quantity independent of the point. This establishes a threshold phenomenon: as the distance to the origin increases beyond 1/2, the minimal marginal density drops to zero, and the size of this drop is independent of the ambient dimension. Joint work with Hermann Koenig.