Maximum of Quadratic form

Question

Let $M$ be a $n \times n$ symmetric matrix. Is it true that $$ \max_{x \in [0,1]^n} |x^T M x| = \max_{y \in \{\pm 1\}^n} |y^T M y|? $$ Clearly, the left hand side is greater than or equal to the right, but I'm not sure if the other inequality holds.

Answer 1

With $n=2$ and: $$M=\begin{pmatrix} 1&0\\0&-1\end{pmatrix}$$ you have: $$y^T M y = y_1^2-y_2^2$$ which is $0$ if $y \in \{-1,1\}^2$ but the max over $[0,1]^n$ is not $0$.