  In this note we prove two results on the quantitative illumination parameter
f(d) of the unit ball of a d-dimensional normed space introduced by K. Bezdek
(1992). The first is that f(d) = O(2^d d^2 log d). The second involves Steiner
minimal trees. Let v(d) be the maximum degree of a vertex, and s(d) of a
Steiner point, in a Steiner minimal tree in a d-dimensional normed space, where
both maxima are over all norms. F. Morgan (1992) conjectured that s(d) <= 2^d,
and D. Cieslik (1990) conjectured v(d) <= 2(2^d-1). We prove that s(d) <= v(d)
<= f(d) which, combined with the above estimate of f(d), improves the
previously best known upper bound v(d) < 3^d.
