Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, and consider the $l\_p$ norm ($p\geq 2$).
Can we prove that the following problems are equivalent:
$$\max\_{\|x\|\_p=\|y\|\_p=1} \left| \langle x, Ay\rangle \right|$$
and 
$$\max\_{\|x\|\_p=1} \left| \langle x, Ax\rangle \right| $$


Can the result be generalized to symmetric tensor and symmetric multilinear form? In particular, if $F: R^n\times\cdots\times R^n$ is a $m$-order symmetric multilinear form, can we prove the problem
$$\max\_{\|x\_1\|\_p=\cdots=\|x\_m\|\_p=1} \left| F(x\_1,\ldots,x\_m) \right|$$
is equivalent to
$$\max\_{\|x\|\_p=1} \left| F(x,\ldots,x) \right| $$
If not, can we prove if for any special case? (the case $p\neq 2$ would be more interesting.)


Thank you very much!


PS: I would even highly appreciate if someone could give me an example where the above statement fails.



Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?
This is false for every $p>2$. Take $A=\left( \begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix} \right)$. Then the maximum of $|\vec{x}^T A \vec{x}|$ on $|x|\_p=1$ is $1$, achieved on the coordinate axes. (The proof is an easy computation with Lagrange multipliers.) But the maximum of $\vec{x}^T A \vec{y}$ is $2^{1-2/p}$ (achieved at $(\sqrt[p]{1/2}, \sqrt[p]{1/2})$, $(\sqrt[p]{1/2}, - \sqrt[p]{1/2})$), which is greater for $p>2$. 
(Of course, the claim is true if $A$ is positive definite. By the Cauchy-Schwartz inequality, if $A$ is positive definite, $|\vec{x}^T A \vec{y}| \leq \sqrt{(\vec{x}^T A \vec{x}) (\vec{y}^T A \vec{y})} \leq \max(\vec{x}^T A \vec{x}, \vec{y}^T A \vec{y})$, so $\max\_{\vec{x}, \vec{y} \in S} |\vec{x}^T A \vec{y} | = \max\_{\vec{z} \in S} \vec{z}^T A \vec{z}$ for any $S$ at all.)


For $p=2$ and $F$ of any degree, this is true and is a theorem of Banach: "[Über homogene Polynome in $(L^2)$](https://eudml.org/doc/218624)" Studia Mathematica (1938) Volume: 7, Issue: 1, page 36-44 . A proof in English can be found as Proposition $1.2^{\circ}$ in [Polynomials and multilinear mappings in topological vector-spaces](https://eudml.org/doc/217555) Jacek Bochnak; Józef Siciak Studia Mathematica (1971) Volume: 39, Issue: 1, page 59-76


I learned all of this from "[Estimates for polynomial norms on $L\_p(\mu)$ spaces](http://dx.doi.org/10.1017/S0305004100064185)", I. Sarantopoulos, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 99, Issue 02, March 1986, pp 263-271, which has much more to say about the problem. 


