Let v_{1} = \left [ \begin{matrix} 1 \\ 0 \end{matrix} \right ] , v_{2} = \left [ \begin{matrix} 2 \\ 3 \end{matrix} \right ] , v_{3} = \left [ \begin{matrix} 5 \\ 4 \end{matrix} \right ] , v_{4} = \left [ \begin{matrix} 3 \\ 0 \end{matrix} \right ] , p = \left [ \begin{matrix} \frac{10}{3} \\ \frac{5}{2} \end{matrix} \right ] , \text{and} S = \{v_{1},v_{2},v_{3},v_{4}\} . Then
\frac{1}{4}v_{1} + \frac{1}{6}v_{2} + \frac{1}{2}v_{3} + \frac{1}{12}v_{4} = p (2)
Use the procedure in the proof of Caratheodory’s Theorem to express p as a convex combination of three points of S.
THEOREM 10 : (Caratheodory)
If S is a nonempty subset of \mathbb{R} ^{n} , then every point in conv S can be expressed as a convex combination of n + 1 or fewer points of S.