I'm reading a paper which states the following inequality, but the (presumably) elementary proof is cited to be in a document, which is too old to get access to.
Let $p: \mathbb{R}^2 \to \mathbb{R}$ be a polynomial of degree $\leq \rho$ and $T \subset \mathbb{R}^2$ a non trivial triangle. We denote with $\mu(T)$ the measure of $T$ and $$ |p|_{k,\infty, T_j} := \max_{|\alpha|=k} \sup_{x \in T} |\partial^\alpha p(x)| $$
Where $\alpha$ is a multi-index.
Now let $\Omega \subset \mathbb{R}^2$ be a polygon and $\Omega = \cup_{i=1}^m T_i$ a decomposition in disjoint (except boundary) triangles. Then there is a constant $c$, such that for all $k = 0,1,2...$, for all $p$ with degree at most $\rho$ and for all decompositions $(T_i)_{i\in I}$ in such triangles (each $I$ finite) with a lower positive bound for their smallest angle holds:
$$ |p|_{k,\infty,T} \leq c\,(\mu(T_j))^{- \frac{1}{2}}(\operatorname{diam}(T_j))^{-k}\|p\|_{L^2(T_j)}\ \text{for all } T_j \in (T_i)_{i\in I} $$
The paper claims this is based on the equivalence of all the norms for a finite dimensional space (the polynomials, most certainly). I failed finding a sufficient norm. Any ideas how to proof this?