12 problems

Golden-rectangle lower-bound conjecture. One might conjecture that

Let $n$ be the number of congruent non-overlapping circles packed in a rectangle, and let $L/S$ denote the rectangle's aspect ratio, where $L$ and $S$ are its longer and shorter si…

Let $\mathcal F$ be the family of bottomless rectangles in the plane, and let $m_k^*(\mathcal F)$ denote the corresponding threshold for decomposing coverings into $k$ coverings. B…

Projected-family piercing conjecture. There exists a constant $A$ such that every finite family of axis-parallel rectangles satisfies

Let $R_{a,1}$ be a rectangle with side lengths $a$ and $1$, let $u_D$ be the torsion function of a domain $D$, and let $\lambda_1(D)$ be its first Dirichlet eigenvalue. Define … Le…

Rectangular eigenvalue-gap conjecture. We conjecture that

Let $R_a$ be a rectangle of unit area, parametrized by its side-length ratio $a\geq 1$, and let $\lambda_1(a)$ denote the first eigenvalue of the Dirichlet biharmonic operator on…

For $b>0$, let $R_b$ be the rectangle whose Neumann eigenvalues are $\mu_{p,q}=\frac{\pi^2p^2}{b^2}+\pi^2q^2b^2$ for $(p,q)\in\mathbb Z_{\geq 0}^2$, ordered as…

Three-rectangle conjecture. In the Main Theorem, the number of rectangles can be reduced to $3$.

Anchored rectangle packing conjecture. Bob can choose an anchored empty rectangle $r(s)$ for every $s\in S$ such that their union covers at least half of $U$.

Let ${\bf R}_{\pi\epsilon,\pi}$ be the rectangle with side lengths $\pi\epsilon$ and $\pi$, and let $\mathcal D$ be a minimal 3-partition that, for $\epsilon>\sqrt{3/8}$, has the s…

Let ${\bf R}_{\pi\epsilon,\pi}$ be the rectangle with side lengths $\pi\epsilon$ and $\pi$, and let $\mathfrak L_3(\epsilon)$ denote its minimal 3-partition energy. For…