for which values of $q$ is $S_q = \{x = (x_1,\dots,x_n) \in \mathbb{R^n}\;|\; \sum_{i} |x_i|^q = 1 \}$ a submanifold of $\mathbb{R^n}$?

Question

I already showed that for $q > 1$ it is a submanifold.

Indeed if you consider $f : \mathbb{R^n} \to \mathbb{R}$ defined by $f(x) = \sum_{i} |x_i|^q - 1 $

then on the one hand we have $f^{-1}({0}) = S_q$

and $df(x) = q(|x_1|^{q-1},\dots,|x_n|^{q-1})$ exists and is continuous.

and because $0 \notin S_q$ then $df(x) \neq 0 \; \forall x \in S_q$ so $S_q$ is a submanifold of $\mathbb{R^n}$ of class $C^1$ and dimension $n-1$

however if say $q =1$ and $n=2$

the set $\{x \in \mathbb{R^2\; | \;}|x_1|+|x_2| = 1\}$ when graphed on a plane looks like this

so because of the four angular points intuitively it shouldn't be a manifold but how would you show that formally ?