Proposition 20

Let $f$ be a convex function, defined over a convex set $X$ of the affine space E. Then for every family $(x_i)_{1 \leq i \leq p}$ of $p$ points of X and every family $(\lambda_i)_{1 \leq i \leq p}$ of $p$ real numbers, all $\geq 0$, such that $\sum_{i=1}^p \lambda_i = 1$, we have:

(3)

If $f$ is strictly convex and if $\lambda_i > 0$ for all $i$, then

(4)

unless all the $x_i$ are equal.

The inequality (3) follows from II, prop. 19 above and II, p. 8, prop. 1. Suppose that the $x_i$ are not all equal (which implies $p \geq 2$) and that the $\lambda_i$ are all $> 0$; then the point $z = \sum_{i=1}^p \lambda_i x_i$ differs from at least one $x_i$. Suppose for definiteness that $z \neq x_1$, write $z = \lambda_1 x_1 + (1 - \lambda_1) y_1$ where $y_1 = \sum_{i=2}^p \frac{\lambda_i}{1 - \lambda_1} x_i$. Then $y_1 \neq x_1$ and, as $0 < \lambda_1 < 1$, we have, by hypothesis,

But by (3) $f(y_1) \leq \sum_{i=2}^p \frac{\lambda_i}{1 - \lambda_1} f(x_i)$, and the inequality (4) follows.