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I am struggling solving the following exercise:

Let $a \in \mathbb{R}^n$ and $ b \in \mathbb{R}$ and define $f : \mathbb{R}^n \rightarrow \mathbb{R}$ by

$f (x)=\langle x,a \rangle + b, x\in \mathbb{R}^n$

Show that for every convex set $X$ in $\mathbb{R}^n$, $ f (X) =\{ f(x):x \in X \} $ is a convex set in $\mathbb{R}$.

Thank you!

Raphael
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  • A convex set in R is an interval. If X is convex then for any $x_0$ and $x_1$ in X the interval of points $x = x_0+t(x_1 - x_0)$ for t in 0 to 1 is in X. Apply f to x and show that it is also an interval. – Paul Sep 24 '14 at 08:47

1 Answers1

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Well, if $t_1+t_2=1$ and $t_1,t_2\geq 0$ then for arbitrary Elements $f(x_1),f(x_2)\in f(X)$ $$ t_1f(x_1)+t_2f(x_2)=t_1\langle x_1,a \rangle+t_1b+t_2\langle x_2,a \rangle+t_2b = \langle t_1x_1+t_2x_2,a \rangle + (t_1+t_2)b $$ and so this is $f(t_1x_1+t_2x_2)$ and (since $X$ is convex) also an Element of $f(X)$

Blah
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