L1 and L0.5 regularisation in machine learning

Question

Why does the function $(x^\frac{1}{2} + y^\frac{1}{2})^2$ is not a convex function? I heard that in machine learning we use L1 regularisation for drop some features because of L1 regularisation has probability to cross zero of function. But L0.5 regularisation can do the same thing and probability will be higher, but we don't use it because L0.5 regularisation is not a convex function that's why loss function will not be convex too. But why L0.5 regularisation is not convex? Thank you in advance

Answer 1

If a function is convex, then $$f\left(\frac12 s + \frac12 t\right) \le \frac12 \left( f(s)+f(t)\right)$$

Now, $f(0,1)=1=f(1,0).$

But their midpoint $f(0.5, 0.5) = (\sqrt{0.5}+\sqrt{0.5})^2=0.5+2(0.5)+0.5>1$. Hence it is not convex.