  We produce new upper and lower bounds for the s-Frobenius number by relating
it to the so called s-covering radius of a certain convex body with respect to
a certain lattice; this generalizes a well-known theorem of R. Kannan for the
classical Frobenius number. Using these bounds, we obtain results on the
average behavior of the s-Frobenius number, extending analogous recent
investigations for the classical Frobenius number by a variety of authors. We
also derive bounds on the s-covering radius, an interesting geometric quantity
in its own right.
