6
votes
1answer
215 views

Has this generalization of a determinant (assigning multiplicities to the rows) been studied?

I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant: Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
10
votes
1answer
361 views

When is a matrix similar to a non-negative matrix?

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 $A'=U^{-1}AU$ is a non-negative ...
2
votes
1answer
122 views

Does the Border (Boundary) Points of a convex body make a concave function?

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 point on the x-axis ...
2
votes
0answers
35 views

Conditions under which a set of points have a low weight representation under some basis

Let $S \subset R^d$ be a convex polytope in $d$ dimensional real space. Say that $S$ has weight $w$ if there exists some basis $x_1,\ldots,x_d$ such that for every point $v \in S$, we can write $v = ...
3
votes
3answers
324 views

Find a convex hull that contains given points?

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 ...
27
votes
4answers
1k views

Probability of zero in a random matrix

Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, ...
4
votes
1answer
373 views

Embed the intersection of an n-dimensional unit $L_1$ sphere and a hyperplane into an (n-1)-dimensional unit $L_1$ sphere.

In $\mathbb{R}^n$, given an unit $L_1$ sphere $\mathcal{B}_n: |x_1|+|x_2|+\ldots+|x_n|\leq 1$ and a hyperplane $\mathcal{P}: a_1x_1+a_2x_2+\ldots+a_nx_n=0$. Does there always exist a rotation such ...
3
votes
2answers
737 views

The space of probability measures and its intersection with hyperplanes in the space of measures

Let $X$ be some uncountable standard Borel space (e.g., the real line). Let $D$ be the set of Borel probability measures on $X$. Let $M$ be the set of signed Borel measures on $X$ Now let ...
3
votes
1answer
330 views

Connections between a polytope's symmetry group and the existence of periodic orbits

Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with ...
1
vote
1answer
371 views

Is the direction of the longest line of a polytope unique?

The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions. The affine subspace is given by: $X \mbox{ u} = y$ where $u$ ...