9 problems

Let $K$ be a Lagrangian submanifold and let $\phi$ range over Hamiltonian isotopies, so that $\phi(K)$ ranges over the Hamiltonian isotopy class of $K$. The monotone Clifford torus…

Berezovik's volume conjecture. For every symplectically self-polar convex body $X\subset\mathbb{R}^{2n}$,

Minimum-volume conjecture. If $n \geq 8$ is even, then the minimum volume is achieved on the antiprism, so that $P_n^{\min}=A(k)$ where $n=2k+2$. If $n\geq 11$ is odd, then the min…

Let $b$ be a prescribed brightness, and consider convex bodies of revolution in three-dimensional space with constant brightness $b$. Campi–Colesanti–Gronchi conjecture. The body c…

Let $b$ be a prescribed brightness, and let $K$ be a convex body in $\mathbb{R}^3$. Its surface area measure has density $s_K$ when … for Borel sets $H\subset\mathcal{S}^2$. Consid…

Let $d$ be a prescribed thickness, and consider convex bodies in three-dimensional space with thickness $d$. The Heil body is the convex hull of six circular arcs of radius $d$, ce…

Lowest-volume conjecture. The unique volume minimizer in $L_3$ is the orbifold $\mathcal{O}_K$.

Let $S^n$ be the unit round sphere, and consider all non-totally-geodesic minimal hypersurfaces in $S^n$. Solomon's conjecture. The least-volume hypersurfaces are given by … if…

Let a generalized Sasakian manifold define a supersymmetric $AdS_5$ solution of type IIB supergravity, and let its generalized Sasakian structure have contact volume. The dual four…