- #1

TyroneTheDino

- 46

- 1

## Homework Statement

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Let X be a vector space over ##\mathbb{R}## and ## f: X \rightarrow \mathbb{R} ## be a convex function and ##g: X \rightarrow \mathbb{R}## be a concave function. Show: The set {##x \in X: f(x) \leq g(x)##} is convex.

## Homework Equations

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If f is convex ##\rightarrow## ##f(\alpha x + (1-\alpha)y) \leq \alpha f(x) + (1-\alpha)f(y)##

If g is concave ##\rightarrow## ##g(\alpha x + (1-\alpha)y) \leq \alpha g(x) + (1-\alpha)g(y)##

##\forall x,y \in X ## and ## \alpha \in \mathbb{R}## where ## 0 < \alpha < 1##

## The Attempt at a Solution

I just assumed that because

##f(x) \leq g(x)## ##\rightarrow## ##f(\alpha x + (1-\alpha)y) \leq g(\alpha x + (1-\alpha)y)##.

And also

##\alpha f(x) + (1-\alpha)f(y) \leq \alpha g(x) + (1-\alpha)g(y)##.

Then because we have that g is convex and f is concave from the relevant equations we have:

## f(\alpha x + (1-\alpha)y) \leq \alpha f(x) + (1-\alpha)f(y) \leq \alpha g(x) + (1-\alpha)g(y) \leq g(\alpha x + (1-\alpha)y)##

I know this isn't enough to show that the set is convex, but I am wondering I am on the right track of thinking and what possible next steps should be.