26 problems

Let $\Omega\subset\subset D$ be convex domains with $C^2$ boundaries in $\mathbb{C}^n$, and let $f$ be continuous on $bD$. Globevnik–Stout conjecture. If $f$ extends continuously t…

Let $P$ be a convex polygonal domain, and let $\Omega$ be a simply-connected smoothly bounded domain. Two domains are Steklov isospectral if their Steklov spectra, counted with mul…

Let $\Omega$ be a bounded convex domain in the plane, let $\lambda_1$ be its first Dirichlet eigenvalue, and let $u_1$ be the corresponding $L^2$-normalized eigenfunction. Define ……

Ba{n}uelos and Burdzy's conjecture. If $\mu_1(\Omega)$ has multiplicity $2$, then

Let $\Omega\subset\mathbb C^N$ be a bounded convex domain whose boundary contains no non-trivial analytic discs. A sequential horosphere is a set of the form … where…

Let $\Omega\subset\mathbb C^N$ be a bounded convex domain, let $z_0\in\Omega$, and let $\mathcal G_{z_0}$ be the family of all geodesic segments and geodesic rays starting from…

Let $\Omega\subset\mathbb C^N$ be a bounded convex domain. It is complex visible if for every distinct $p,q\in\partial\Omega$ there exist a compact set $K\Subset\Omega$ and open ne…

The cluster-set conjecture. If $p,q\in\Gamma(\{\gamma_k\})$ and $p\ne q$, then either

Pólya functional bounds conjecture. For every bounded convex planar domain $D$,

Let $\Omega\subset\mathbb{R}^2$ be a bounded, open, convex set, and let … where $\mu_1(\Omega)$ and $\sigma_1(\Omega)$ are respectively the first nonzero Neumann and Steklov eigenv…

Let $\Omega$ be a convex domain in $\mathbb{R}^2$, let $L(\Omega)$ denote its perimeter, and let $\mu_2(\Omega)$ be its first nonzero Neumann eigenvalue. Laugesen's conjecture. … E…

Let $\Omega$ be a convex domain in $\mathbb{R}^2$ such that $\kappa_1=\mu_2$. Here $\kappa_1$ denotes the first eigenvalue of the variational problem defined in the paper, $\mu_2$…

Fundamental gap conjecture. The fundamental gap is at least $3\pi^2/D^2$.

Cube-normalized capacity uniqueness conjecture. All cube-normalized symplectic capacities coincide on $X$.

Let $d\geq 1$, let $C\subset\operatorname{\mathbb{R}}^d$ be a convex domain, and let … be its tube domain. Assume that $(\Omega,K_\Omega)$ is Gromov hyperbolic; equivalently, $C$ i…

Let $d\geq 1$, let $C\subset\operatorname{\mathbb{R}}^d$ be a bounded convex domain, and let $\Omega=C+i\operatorname{\mathbb{R}}^d$ be its tube domain. Bounded-base tube-domain co…

Let $\Omega\subset\mathbb{R}^n$ be a bounded convex domain, and let $u$ be a first Dirichlet eigenfunction of $\Omega$. Write $\operatorname{diam}(\Omega)$ for its diameter and…

Continuous extension conjecture. Every complex geodesic of $\Omega$ extends continuously to $\overline{\mathbb{D}}$.

Let $1<\alpha<2$ and let ${\boldsymbol{x}}\in B^2_\alpha$. Define … where ${\boldsymbol{u}}$ satisfies ${\boldsymbol{x}}+\delta{\boldsymbol{u}}\in\partial D$ and ${\boldsymbol{v}}$…

Let $K\Subset\mathbb{C}$ be a compact convex domain with nonempty interior, let ${\mathcal P}_n(K)$ denote the polynomials of degree at most $n$ on $K$, and let…

Equality characterization conjecture. $F_{\Omega}(w)=1$ if and only if there exists a balanced domain $\Omega'$ (not necessarily convex) and a biholomorphic mapping $H:OmegatoOmega'$…

Let $\Omega\subset\mathbb{R}^n$ be a convex domain, and assume $n>1$. Write $\xi(\Omega)$ for the gap function, namely the difference between the first two Dirichlet eigenvalues of…

Let $\Omega$ be a convex domain of given diameter $D$, and let $0=\mu_1(\Omega)<\mu_2(\Omega)\leq\mu_3(\Omega)\leq\cdots$ be the Neumann Laplacian eigenvalues. Disk minimization co…

Fundamental gap conjecture. The eigenvalues satisfy the inequality above.

Let $D_1,D_2\subset \mathbb{R}^d$ be smooth bounded convex domains, with $d\geq 1$. For $i=1,2$, let $p_{D_i}(t,x,y)$ denote the Neumann heat kernel in $D_i$. If $D_2\subset D_1$,…