# Understand a definition

Reading Randomized Rounding without Solving the Linear Program paper I came across this definition:

Let $$P$$ be a convex set in $$\mathbb{R}^n$$ and let $$f$$ be a linear function (not necessarily homogeneous) from $$P$$ to $$\mathbb{R}^m$$.

Correct me if I am wrong, but $$f$$ is a non-homogenous function that takes a set of points in $$\mathbb{R}^n$$ and returns a set of points in $$\mathbb{R}^m$$?

A homogeneous function $$f$$ satisfies $$f(ax)=af(x)$$ for scalars.
When $$f(x)=Ax+b$$ is linear, this is to say that $$b=0$$. So a non-homogeneous linear function may have a non-zero constant vector term.
So in your case, $$f$$ may or may not be homogeneous/non-homogeneous. Strictly speaking it doesn't map $$\mathbb{R}^n\to\mathbb{R}^m$$, it maps $$P\to\mathbb{R}^m$$. But since $$f$$ is linear, the extension is a natural one, and it makes good sense to just think of it as mapping $$\mathbb{R}^n\to\mathbb{R}^m$$.