example of sub-harmonic function

Question

A continuous function $u:\mathbb{R}^n\to\mathbb{R}$ is sub-harmonic if for every $x_0\in\mathbb{R}^n$ and $r>0$ $$u(x_0) \leq \frac{1}{|\partial B(x_0,r)|}\int_{\partial B(x_0,r)} \!\!u(x)\ d\sigma(x)\ .$$

Using only this definitions, how could I prove that $x\mapsto|x|$ is sub-harmonic?

Answer 1

Hint: Use the Triangle Inequality to show $\left|\frac{x+y}{2}\right|\le\frac12|x|+\frac12|y|$ and integrate over a sphere.

Hint 2: Integrate $|x_0|\le\frac12|x|+\frac12|2x_0-x|$ over the sphere centered at $x_0$ noting that $2x_0-x$ is the point opposite $x$.

Answer 2

You have to prove that $|x_0| \le \frac{1}{|\partial B(x_0,r)|} \int_{\partial B(x_0,r)} |x|dx$.

Now, $x_0 = \frac{1}{|\partial B|}\int_{\partial B} x_0 = \frac{1}{|\partial B|}\int_{\partial B} x \le \frac{1}{|\partial B|}\int_{\partial B} |x|$. The last term is positive, therefore taking the modules the result follows.