Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 7H / a

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Solution

For

0 \leq t \leq 1

and

x, y \in R^{d}

, convexity gives

f_{i} (t x + (1 - t) y) \leq t f_{i} (x) + (1 - t) f_{i} (y) .

(1)

Taking the maximum over

i

and then bounding each term by the corresponding endpoint maxima yields

max_{i} f_{i} (t x + (1 - t) y) \leq t max_{i} f_{i} (x) + (1 - t) max_{i} f_{i} (y) .

(2)

Thus the finite pointwise maximum of convex functions is convex. Summing the original inequalities over $i$ proves that $\sum_{i} f_{i}$ is convex as well.

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7H
Paper 1
ib
2021
Past exam of the mathematics course of the University of Cambridge
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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 7H / a

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