Let C be a convex
compact set in a normed space E and let skel1C be the subset of C that
contains those boundary points of C which are not centres of 2-dimensional
balls in C. When l is a continuous functional on E, we say that the path
P = g([α,β]) is l-strictly increasing if l(g(t1)) < l(g(t2)) for every t1, t2 such
that α ≤ t1< t2≤ β. D. G. Larman proved the existence of an l-strictly
increasing path on the one skeleton of C with l(g(α)) =minx∈Cl(x) and
l(g(β)) =maxx∈Cl(x).
In this paper we prove a theorem concerning the number of l-strictly
increasing paths on the one-skeleton of C, that are mutually disjoint and along
each of which l assumes values in a range arbitrarily close to its range on
C.
