Questions related to the Mahler and the Viterbo Conjecture
A question that I'm very much interested in is: for \( K\subset \mathbb{R}^n \) a centrally symmetric convex body with polar dual body \( K^{\circ}\subset \mathbb{R}^n \) and \(c\) any normalized symplectic capacity, is it true that \(\frac{c(K\times K^{\circ})^n}{n!}\leq Vol(K\times K^{\circ})\) holds? This is part of the Viterbo isoperimetric conjecture and it is known ( by a result of Arstein-Avidan, Karasev and Ostrover ) to imply the Mahler volume conjecture. While the original and more general statement of the Viterbo conjecture has now been proven wrong ( by Haim-Kislev and Ostrover ), there's still hope that the less general case stated above does hold. Here's a list of related questions in this topic: Is it true that \(c_{Gr}(K\times K^{\circ})=4\) for any \(K\) and \(c_{Gr}\) the Gromov width? This immediately implies the Mahler conjecture, since the Viterbo conjecture is easily seen to hold for the Gromov width. This has been shown to be true for \(K\) being...