4 problems

Let $d\geq 2$, let $S\subseteq\mathbb{R}^d$ be a set whose $(d-1)$-st Beer index $\operatorname{b}_{d-1}(S)$ is defined, and assume that every p-component of $S$ is contractible. L…

For a set $X\subseteq\mathbb{R}^d$ and an integer $k\geq 0$, define … where $\binom{X}{k+1}$ is the set of $(k+1)$-element subsets of $X$. Let $\operatorname{b}_k(S)$ be the $k$-th…

Let $S\subseteq\mathbb{R}^2$ be a measurable set, and let $\operatorname{b}(S)$ denote its Beer index and $\lambda$ planar Lebesgue measure. Boundary convexity conjecture. For ever…

Call a set $S\subseteq\mathbb{R}^2$ p-componentwise simply $\Delta$-connected if, for every triangle $T$ with $\partial T\subseteq S$, one has $T\subseteq S$. Assume that…