Take a polynomial $g\in\mathbb{R}[\mathbf{x}]$, in $n$ variables and having some degree $d$, with $g(\mathbf{x})\geq 0$. We define the $p$-norms of $g$ as $$ \vert \vert g \vert \vert _{p} = \left( \int_{\mathbb{S}^{n-1}} \vert g \vert^{p} d\mu \right)^{\frac{1}{p}}. $$ Given the finite dimension of the space, there are constants $A,B>0$ such that $$ A\vert\vert g\vert\vert_{2} \leq \vert\vert g\vert\vert_{1} \leq B\vert\vert g\vert\vert_{2}. $$
Is it possible to change restrict/modify one of the constants given that the other can change freely?
For example, if you know $A=\frac{1}{2}$, is it possible to enforce $A>1$? (you have the freedom to adjust $B$ as needed).