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 and the supporting function in .
In the epigraph we are going to have a hyperplane of n+1 dimensions and a hyperplane of n dimensions for . So what I"m thinking is to find find the maximum slope of of the function 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 but still remain above . Now every value from will be below the hyperplane (from the epigraph) showing that is convex.