## convexity and epigraph question

Prove that if the epigraph is a convex set then the function below the epigraph is convex.

Here's what I have so far.

The epigraph is going to be in $\displaystyle \mathbb{R}^{n+1}$ and the supporting function in $\displaystyle \mathbb{R}^n$.

In the epigraph we are going to have a hyperplane of n+1 dimensions and a hyperplane of n dimensions for $\displaystyle f(x_1,x_2,..x_n)$. So what I"m thinking is to find find the maximum slope of of the function $\displaystyle f$ and rotating the hyperplane in the epigraph so that both are going to be parallel (this is the part I have no idea on how to do, as we can possibly in 4+ dimensions. I have no idea how a rotation matrix will look like). At which point we simply lower the hyperplane from the epigraph to hit the function $\displaystyle f(x_1,x_2,..x_n)$ but still remain above $\displaystyle f(x_1,x_2,..x_n)$. Now every value from $\displaystyle f(x_1,x_2,..x_n)$ will be below the hyperplane (from the epigraph) showing that $\displaystyle f(x_1,x_2,..x_n)$ is convex.