Theorem 13.1.6 (Convex Criteria).label Let $c\suf\R^{n}$ then the following are equivalent
- (1)
$c$ is convex
- (2)
$\fall m\geq 2\in\Z,\curl{x_i}_{1}^{m}\in c$ and for each $\lam_{1},...,\lam_{m}\geq0$ such that $\sums{m }{i=1}\lam_{i}=1$ we have $\sums{m }{i=1}\lam_{i}x_{i}\in C$
Proof. The proof is by induction on $m$ with base case given by convexity. Let $\curl{x_i }_{1}^{m+1}\in c$, $\lam_{1},...,\lam_{m+1}\geq 0$ such that $\sums{m+1 }{i=1}\lam_{i}=1$. WLOG $\lam_{i}\nin \curl{0,1}$ for each $i$. Observe $s\define \sums{m }{i=1}\lam_{i}\in (0,1)$ so
\begin{align*}\sums{m+1 }{i=1}\lam_{i}x_{i}=s \sums{m }{i=1}\frac{\lam_{i}}{s}x_{i}+\lam_{m+1}x_{m+1}\end{align*}
Since $\sums{m }{i=1}\frac{\lam_{i}}{s}=1$, by induction hypothesis $s \sums{m }{i=1}\frac{\lam_{i}}{s}x_{i}\in c$ hence $\sums{m+1 }{i=1}\lam_{i}x_{i}\in c$ as well.$\square$
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