The "$p$-norm" with $0<p<1$ is quasiconcave for nonnegative vectors

Question

This question is related to my previous one, Is the "$p$-norm" with $0<p<1$ a concave function.

We have that $\|x\|_p= (\sum_i |x_i|^p)^{1/p}$ is not concave in general. Can we say though that it is quasiconcave for nonnegative vectors? Namely if $x,y\in \mathbb{R}^n$ have nonnegative coordinates, then do we have $$\|x+y\|_p\geq \min(\|x\|_p,\|y\|_p).$$ Any hint on how to prove this?

Answer 1

Even stronger: $\|x+y\|_p\geq \max(\|x\|_p,\|y\|_p)$ for nonnegative $x, y$.

Indeed, without loss of generality $\max(\|x\|_p,\|y\|_p) = \|x\|$. Since $x_i,y_i\ge 0$ for all $i$, we have $|x_i+y_i| \ge |x_i|$, hence $\|x+y\|_p\geq \|x\|_p$.