Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees

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.