Where do homogeneous polynomials on \(\ell_{1}^{n}\) attain their norm? (Q1826871)

In 2001, Zalduendo conjectured that almost all homogeneous polynomials on \(\ell_\infty^n\) will attain their norms at vertices of the unit ball when \(n\) tends to infinity. In this article, the authors show that this conjecture fails for \(\ell_1^n\). Namely, using the normalised Lebesgue measure \(\mu_n\) on the unit ball of the space of symmetric bilinear forms on \(\ell_1^n\), it is shown that for \(C=\{P \in {\mathcal P}(^2 \ell_1^n)\) such that \(\| A\|=1\) and \(P\) does not attain its norm at a vertex\(\}\), we have \(\mu_n(C) \geq 1/6\) (here \(A\) is the symmetric bilinear form associated with \(P\) by the polarisation formula). However, in the main result of the article, it is shown that most of the polynomials \(P \in {\mathcal P}(^2 \ell_1^n)\) attain their norm, in the low dimensional faces of the unit ball of \(\ell_1^n\). If \(S_n^m=\{P \in {\mathcal P}(^2 \ell_1^n)\) such that \(\|A\|=1\) and \(P\) attains its norm in the interior of a \((m-1)\)-face\(\}\), then \(\lim_{n \rightarrow \infty} \mu_n (\bigcup_{m>16\sqrt n} S_n^m) = 0\).