Recently I came across a statement in a paper that I am unable to verify. Namely, it roughly says that the oscillation of a polynomial on a cube can be controlled by the oscillation of the polynomial of a slightly smaller subcube.
More precisely; let $P$ be a nonnegative polynomial in $n$ variables and of total degree at most $d$. Let $Q_0$ be any cube in $\mathbb{R}^n$ and let $Q_1$ be a child of cube $Q_0$ (i.e. one of the cubes that we get when we partition $Q_0$ into $2^n$ equal cubes).
Assume that $$ \max_{Q_0} V - \min_{Q_0} V > C \text{diam}(Q_0)^{-2} $$ and $$ \max_{Q_1} V - \min_{Q_1} V \leq C \text{diam}(Q_1)^{-2}.$$ Does there exist a constant $c>0$ depending only on $n,d$ (not on the cube or the polynomial) such that we have $$ cC(\text{diam}(Q_1))^{-2} \leq \max_{Q_1} V - \min_{Q_1} V. $$
Here $C>0$ is some constant that I could also make as large as needed depending only on $n,d$ if needed.
I do understand how to compare $\max_{Q_0} V$ and $\max_{Q_1} V$ (one can simply consider the linear map $P\vert_{Q_0} \mapsto P\vert_{Q_1}$. The space of polynomials in $n$ variables and degree at most $d$ are finite dimensional, thus, the map is continuous. If we put the norms $\max_{Q_0} \vert P\vert$, respectively $\max_{Q_1} \vert P \vert$, then we get the desired statement). The issue is that I have no handle on the minimum.
