Consider a real square matrix $A$ of size $n\times n$. Under which conditions on $A$ does there exist a row-stochastic matrix $U$ (non-negative, rowsums = 1), such that $A'=U^{-1}A …

Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which …

Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$? I found only one reference …

Is there any test to tell me whether a straight line in a 3D euclidean space passes through a bounded closed convex region? To focus on a more specialised version of the problem, y …

I have recently become aware of the following neat statement. Consider a convex polygon $P$ in the real plane with integral vertices. If we associate with every integral point $(a …

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas? Does every convex polyhedron have a combinatorially isomorphic coun …

An edge of a triangulated manifold is said to be contractible if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the …

Let $\mathbb{S}$ be a closed and bounded convex body in 2-D with some non-empty intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most poin …

Is the secondary polytope of a simplicial polytope necessarily simplicial?

One has an $n$-dimensional convex polytope $P$ represented by an intersection of half-spaces: \begin{equation}H_i = { (x_1,x_2, \ldots,x_n) \in \mathbb{R}^n \mid \sum_{j=1}^n a_{i …

Consider a set of $n$ points $x_1,\ldots,x_n \in \mathbb{R}^d$, for some $n \gg d$. Suppose ${x_1,\ldots,x_n} \subset C \subset \mathbb{R}^d$. Say that a set of points $y_1,\ldots, …

Given a weighted, undirected graph G with K knodes k1 … kK. I have a K times K matrix containing the shortest distances between each pair of points. Is it possible (if yes how), t …

Hi, overflowers. I was interested in a sharp lower bound for the number of lattice points (say, integral lattice points) inside the tetrahedron defined by the coordinate hyperpla …

I have a linear programming problem where the polytope is described as a convex hull of an exponential number ($2^d$) of points in d-dimensional space. What could be an approach? D …

Suppose that you have vectors $a_{1},...,a_{m}$ in $\mathbb{R}^n$. Can you take $n$ among them, let $v_{1},...,v_{n}$ such that $a_{1},...,a_{m}$ belong to the convex hull of $(cn) …