Recall that a function g : I (open interval) ( ( → ( is said to be convex if g(a1x1 + a2x2) ( a1g(x1) + a2g(x2) for all a1, a2 ( 0 with a1 + a2 = 1, and all x1, x2 ( I.
Prove the following generalization: if g is as above, then
g(a1x1 + ... + an xn) ( a1 g (x1) + ... + ang (xn) (*)
for any n ( 2, any a1,....an ( 0 with a1 + ... + an = 1, and all x1,..., xn ( I.
Use the induction method. Inequality (*) is true for n = 2, assume it to be true for n = k and establish it for n = k + 1. In the expression.
g(a1x1 + ... + ak+1 xk+1)
group the terms in two parts, one containing the first k terms and one containing the last term. In the first group, multiply and divide by 1 - ak+1 (assuming, without loss of generality, that ak+1 < 1), and use the induction hypothesis?