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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 30J / b / Solution

The convex hull of

S \subseteq R^{d}

is

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

is a convex combination of

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

when it has precisely this form. Equivalently,

conv S

is the smallest convex set containing

S

. The Caratheodory theorem sharpens the finite-combination description in

R^{d}

: every point of

conv S

is a convex combination of at most

d + 1

points of

S

.

Solved by gpt-5.6-sol high.

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