TITLE: Is the set $\{(x,y)\in\mathbb R^2:x^{2/3}+y^{2/3}\le1\}$ connected?
QUESTION [7 upvotes]: Is the set $\{(x,y)\in\mathbb R^2:x^{2/3}+y^{2/3}\le1\}$ connected? 
Please help me. I'm clueless.
Added: Is the set convex?

REPLY [2 votes]: Hint:
$B_{2/3}\stackrel{\rm{}def}{=}\{(x,y)\in\mathbb R^2\mid x^{2/3}+y^{2/3}\le 1\}$ is equivalently the set 
\begin{align*}
B_{2/3} &= \{(x,y)\in\mathbb R^2\mid |x|^{2/3}+|y|^{2/3}\le 1\} = \{ {x}\in\mathbb{R}^2 \mid \lVert {x} \rVert_{2/3}^{2/3} \leq 1 \} \\
&= \{ {x}\in\mathbb{R}^2 \mid \lVert {x} \rVert_{2/3} \leq 1 \}
\end{align*}
that is the unit ball for a $p$-norm (where $p\stackrel{\rm{}def}{=}\frac{2}{3}$). This should settle the problem of whether it is connected.
PS: you can see what it looks like on this page.