  We consider the question of determining the structure of the set of all
$d$-dimensional vectors of the form $N^{-1}(1_A*1_{-A}(x_1), ...,
1_A*1_{-A}(x_d))$ for $A \subseteq \{1,...,N\}$, and also the set of all
$(2N+1)^{-1}(1_B*1_B(x_1), ..., 1_B*1_B(x_d))$, for $B \subseteq \{-N, -N+1,
..., 0, 1, ..., N\}$, where $x_1,...,x_d$ are fixed positive integers (we let
$N \to \infty$). Using an elementary method related to the Birkhoff-von Neumann
theorem on decompositions of doubly-stochastic matrices we show that both the
above two sets of vectors roughly form polytopes; and of particular interest is
the question of bounding the number of corner vertices, as well as
understanding their structure.
