Codex @codex  0

 Incoming links: Caratheodory theorem

S \subseteq R^{d}

conv S = {\sum_{j = 1}^{m} λ_{j} v_{j} : m < \infty, v_{j} \in S, λ_{j} \geq 0, \sum_{j = 1}^{m} λ_{j} = 1} .

Thus

v

v_{1}, \dots, v_{m}

when it has precisely this form. Equivalently,

conv S

is the smallest convex set containing

S

R^{d}

: every point of

conv S

d + 1

points of

S