I'm reading a paper about mathematical physics at the moment and am wondering about the following:
Let $w\colon\mathbb{R}^2\to\mathbb{R}$ be defined by $w(x)=-\log|x|$ and $\mu\colon\mathbb{R}^2\to\mathbb{R}$ be some non-negative integrable function radially symmetric about $x_0\in\mathbb{R}^2$ such that $\int\mu(x)~dx=1$. Then by Newton's theorem $$(\mu\ast w)(x)\leq w(x)\text{ for a.e. }x.$$
What are they referring to?
Here $\mu$ is the density of a planet in a two-dimensional universe. $w*\mu$ is the gravitational potential caused by this mass. Whereas $w$ is the gravitational attraction if all the mass of the planet were concentrated at the center of the planet. It is known that if you are at a certain place in the universe, then the only part of the planet which has a non-zero net gravitational attraction upon you is precisely that part of the planet contained in a ball whose radius is your distance from the center of the planet.
For example, if the planet were a perfect hollow sphere (or in the 2D world, a perfect hollow circle), then anyone inside the sphere/circle would feel no gravitational attraction. Whereas anyone outside the sphere/circle would feel the attraction as if the whole mass of the planet were concentrated at one point at the center.