Prove that $L_p$-norm is non-convex, $p < 1$, $p \not = 0$.

Question

How do I show that $$f (x) = \left(\sum_{i=1}^n x_i^p\right)^\frac{1}{p},\; p < 1,\; p \not = 0$$ is not convex? The hint is to use Cauchy-Schwarz inequality: $$\langle a, b\rangle \leq \|a\|_2\|b\|_2 .$$

Answer 1

Pick concrete values for points $x,y\in\mathbb R^n$ and a scalar $\lambda \in [0,1]$. Check if the definition of convexity holds for these values (this requires you to calculate $f(x),f(y),f(\lambda x+(1-\lambda) y)$). If this does not result in a contradiction to the definition of convexity, try other values and repeat.