= Convex combinations in a triangle
{title2=$\lambda_1=1-x/6-y/3,\quad\lambda_2=x/6,\quad\lambda_3=y/3$}
Every allocation $(x,y)$ in the <convex feasible region for a resource budget> is a convex combination of its three <extreme points>:
$$
(x,y)=\lambda_1(0,0)+\lambda_2(6,0)+\lambda_3(0,3),\qquad
\lambda_2=\frac x6,\quad\lambda_3=\frac y3,\quad\lambda_1=1-\frac x6-\frac y3.
$$
The condition $x+2y\leq6$ is exactly $\lambda_1\geq0$; nonnegative $x,y$ give the other two nonnegative <coefficients>. Thus the <half-space> description and the <convex hull> description encode the same <triangle>. At $(2,1)$ all three <coefficients> are $1/3$, and at $(3,9/8)$ they are $1/8$, $1/2$, $3/8$. The green and orange points in the budget diagram therefore have two complementary interpretations: direct resource constraints and mixtures of extreme choices.
The three <extreme points> alone are not a convex set, because their <midpoints> are absent. Their <convex hull> fills the whole <triangle>. More generally, <Caratheodory theorem> bounds how many generating points are needed for one convex combination in a finite-dimensional real <vector space>; its geometric content is that a complicated collection can describe each attainable average with a short representation.
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