Turán numbers of r-graphs on r+1 vertices

Let H_k^r be an r-uniform hypergraph with r+1 vertices and k edges where 3 ≤ k ≤ r+1.
It is easy to see that such a hypergraph is unique up to isomorphism.
The upper bound on its Turán density is (k-2)/r.
In the case k=3, Frankl and Füredi (1984) used a geometric construction to prove lower bound 2^{1-r}.
We use classical results from order statistics going back to Rényi (1953) and a geometric construction to prove a lower bound of order r^{-(1+1/(k-2))}.