6 problems

Let $B$ be the Euclidean unit disk, and for each even integer $n\geq 4$ let $P_n^*$ be the regular polygon used as the anisotropy in the source. Write … where $h_{P_n^*}(B)$ is the…

Cheeger lower-bound conjecture. There exists an absolute constant $c$ such that, for all $\varepsilon\in(0,1)$, there are constants $N(\varepsilon)$ and $M(\varepsilon)$ such that,…

Let $G_2(1)$ be the indicated two-vertex graph, let $\sqcup$ denote disjoint union, and let a staircase graph mean a graph of the staircase type defined in the paper. Staircase rep…

Let $G$ be a Cheeger graph, meaning a graph attaining the first Cheeger constant of the relevant simplex. Bipartiteness conjecture. All Cheeger graphs are bipartite. This is propos…

Let $G$ be a Cheeger graph, meaning a graph attaining the first Cheeger constant of the relevant simplex. Triangle-freeness conjecture. All Cheeger graphs are triangle-free. This i…

For a simplex with $n$ vertices, let $h(n)$ denote its first Cheeger constant. Power-of-two lower-bound conjecture. We have … for all $n=2^\alpha$, with $\alpha\geq 1$. The conject…