17 problems

Let $n\ge3$ be odd, and let $G$ be a complete geometric graph on $n$ vertices that can be decomposed into $\left\lceil\frac{n}{2}\right\rceil+1$ plane star-forests. Convex-hull-siz…

Let $L_n$ be the perimeter of the convex hull of the multiple random walks, and let ${\mathcal G}_n$ and ${\mathcal E}^{\mathrm{f}}_n$ be the sets defined in the paper. For each re…

Asada et al.'s conjecture. A thrackle of convex sets on a point set $P$ of size $n$ has at most $n$ convex hulls.

Agoston et al.'s conjecture. A convex hull thrackle on $n$ points has at most $n$ convex hulls.

Faster-layer computation conjecture. There exists an algorithm to find the rectilinear convex layers of $P$ in better than $O(kn\log n)$ time and $O(n)$ space.

Fix $d\in\mathbb{N}$ and $k\in\{1,\ldots,d\}$. Let $C_{n-1,d}$ be the convex hull object from the paper, and let $f_{k-1}(C_{n-1,d})$ denote its number of $(k-1)$-dimensional faces…

Locally valid-cut conjecture. The intersection of $\mathcal{R}_{\mathrm{shor}} \cap \mathcal{R}_{\mathrm{ksoc}}$ with the locally valid cuts captures $\mathcal{G}$.

Consider an arrangement of a growing number of random lines in $\mathbb{R}^2$, and let the vertex set be the set of all pairwise intersection points of the lines. Let $f_0$ denote…

Let $x\in\mathbb{R}^2$, $X\in\mathbb{R}^{2\times 2}$, $y\in\mathbb{R}^2$, and let $Y_{12},\alpha\in\mathbb{R}$. Let $\mathcal{H}'$ denote the relevant relaxation in the paper, and…

Let $n_1,n_2\in\mathbb{Z}_{+}$, and let ${\mathcal{U}}^{\operatorname{row}}_{(n_1,n_2)}(l,u)$ and ${\mathcal{U}}^{\operatorname{col}}_{(n_1,n_2)}(l',u')$ denote the rank-one nonneg…

Let $X$ be the mixed-integer multilinear feasible region considered in the formulation, and let $\mathfrak{F}^{\lambda}$ be its extended formulation in the variables consisting of…

The logarithmic variance conjecture. Under these assumptions,

Polynomial stabilization conjecture. The iterative construction stabilizes after a polynomially bounded number of iterations:

Let $L_n$ denote the perimeter of the convex hull of the first $n$ steps of a planar random walk, and suppose the degenerate case holds in which … with probability $1$, so that the…

Entropy-hull conjecture. Then

Let $n$ independent Brownian paths in the plane have the same fixed, finite duration and start from the same point, and let $\mathcal{C}_n$ be their global convex hull, the smalles…

Let $\Omega\subset\hat{\mathbb{C}}$ be a hyperbolic domain, and let $r:\Omega\to{\rm Dome}(\Omega)$ be the nearest point retraction from $\Omega$ to the boundary of the convex hull…