Let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$, where $d, m \in\mathbb{N}$ are fixed.

(I.e., the above sum ranges over all multiindices $\alpha=(i_1, \ldots, i_d)\in\mathbb{N}_0^{\times d}$ of length $|\alpha|\equiv i_1+\ldots + i_d$ less than $m$.)

Denote by $P_k = \sum_{|\alpha|=m}c_\alpha\cdot x^\alpha$, $ 0\leq k\leq m$, the $k$-th homogeneous component of $P$.

I was wondering the following: Given a compact subset $K$ in $\mathbb{R}^d$, is it possible to for $\varphi_K(P):=\max_{0\leq k\leq m}\|P_k\|_{\infty; K}$ (or indeed for any $\ell_p$-norm of $(\|P_k\|_{\infty;K})_{k\geq 0}$ with $1\leq p \leq \infty$) find a constant $\kappa=\kappa(d,K)$ such that

$$\tag{1} \varphi_K(P) \ \leq \ \kappa\cdot \|P\|_{\infty; K} \qquad \text{ for each } \ P \ \text{ as above} \ ?$$

(Here, $\|f\|_{\infty;K}:=\sup_{x\in K}|f(x)|$ is the uniform norm over $K$.) Any references are welcome.