MathDB

Let

n

be a positive integer and let

f_1(x), f_2(x) \dots f_n(x)

be affine functions from

\mathbb{R}

\mathbb{R}

such that, amongst the graph of these functions, no two are parallel and no three are concurrent. Let

S

be the set of all convex functions

g(x)

from

\mathbb{R}

\mathbb{R}

such that for each

x \in \mathbb{R}

, there exists

i

such that

g(x) = f_i(x)

.
Determine the largest and smallest possible values of

|S|

in terms of

n

.
(A function

f(x)

is affine if it is of form

f(x) = ax + b

for some

a, b \in \mathbb{R}

. A function

g(x)

is convex if

g(\lambda x + (1 - \lambda) y) \leq \lambda g(x) + (1-\lambda)g(y)

for all

x, y \in \mathbb{R}

and

0 \leq \lambda \leq 1

)

Problem Statement