Maximal sections of the unit ball of $l^n_p(\mathbb{C})$ for $p > 2$
Jacek Jakimiuk, Hermann König
TL;DR
The paper extends Ball’s hyperplane-slicing resilience to the complex setting of the unit balls in $\ell_p^n(\mathbb{C})$ by analyzing the complex polydisc and introducing the normalized section volume $A_{n,p}(a)$. It develops probabilistic and harmonic-analytic representations of $A_{n,p}(a)$, notably a central-limit-theorem-based formula and a Hankel-transform form, to study the asymptotics and regime-dependent behavior. The authors prove a dichotomy: for each $p>2$ there exists a threshold $N(p)$ such that for $n\ge N(p)$ the complex analogue of Ball’s resilience holds, while for large $n$ (and large $p$) resilience fails, with explicit bounds such as $N(p)=\frac{5}{2}p$ for $p\ge9$ and $N(p)=p$ for $p\ge140$, and a secondary regime where $n\le cp$ with huge $p$ preserves resilience. The results illuminate the delicate interplay between dimension and $p$ in the complex setting and clarify where the complex analogue of polydisc slicing behaves similarly to or diverges from the real case. The work thus delineates the parameter regions where maximal complex hyperplane sections align with the diagonal, advancing our understanding of high-dimensional complex convex geometry.
Abstract
Eskenazis, Nayar and Tkocz have shown recently some resilience of Ball's celebrated cube slicing theorem, namely its analogue in $l^n_p$ for large $p$. We show that the complex analogue, i.e. resilience of the polydisc slicing theorem proven by Oleszkiewicz and Pelczyński, holds for large $p$ and small $n$, but does not hold for any $p > 2$ and large $n$.