STOCHASTIC

CHOOSING INVESTMENT PORTFOLIOS
Hence Xf(x l ) + (1 -A)/(x 2 ) >f{Xx l + (1 -A)* 2 }.
Thus/is strictly convex unless a and b are proportional. But this requires
that there exist constants qx and q2 not both zero such that q^a = q2b or
that
qlXSixi l = q2{\-X)Sixi 2 => qiW = 92(1-A)xj 2
(since St > 0)
or that x, 1 and x, 2 are proportional since qx' = q^X and q2' = q2(l—X) are
both not zero. However, the condition that £"=i x, 1 and £?=1 x; 2 = 1 means
that x 1 and x 2 cannot be proportional unless x, 1 = x; 2 for all i, which is a
contradiction.
Remark f is convex but not strictly convex on M = {x \ x ^ 0} because /
is linear on every ray that contains the origin.
Lemma 2 q> is a concave function of /? > /?t, where pL > 0 is defined below,
if 2^ 1 and S, ^ 0, i= l,...,n.
Proof Let Kfi = {& \ e'x = 1, x ^ 0, /(x) ^ /?}. Let pL> 0 be the smallest
0 for which A^ ^ 0. Clearly J?L ^