All Questions
8 questions
6
votes
2
answers
539
views
Conditions for including cones
Consider $N$ $n$-dimensional vectors, where the angle between any two vectors is acute and their starting point is at the origin. Can we rotate these vectors together so that the coordinate components ...
- 61
5
votes
5
answers
1k
views
Finding an axis-aligned ellipsoid of minimal volume which contains a given ellipsoid
A friend asked me to post the following question. He's not an MO user and felt it would be better received if asked by someone who was already known to the community. This is not my area, but I'll do ...
- 30.3k
5
votes
3
answers
1k
views
Proof that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?
Max Koecher (for example, in The Minnesota Notes on Jordan Algebras and Their Applications; new edition: Springer Lecture Notes in Mathematics, number 1710, 1999), defined a domain of positivity for a ...
5
votes
2
answers
134
views
Is there a non-orthogonal linear deformation of a polytope that preserves edge-lengths and vertex-origin-distances?
Is there a polytope $P\subset\Bbb R^d$ (convex hull of finitely many points, not contained in a proper affine subspace), and a linear, but non-orthogonal transformation $T\in\mathrm{GL}(\Bbb R^d)\...
- 13.6k
4
votes
2
answers
492
views
The Aleksandrov-Fenchel inequality of mixed discriminants for Hermitian matrices
Suppose $A,A_1,\ldots,A_{n-2}$ (resp. $B$) are (resp. is) real positive-definite (resp. arbitrary) symmetric $n\times n$ matrices and denote by $D(\cdot,\ldots,\cdot)$ the mixed discriminant. We have ...
- 593
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
1
vote
1
answer
132
views
Can I find $n$ points on the boundary of an $n$-dimensional ball with certain properties?
My problem is the following: I want to construct $n$ rays all starting at a point $v$ that is not in the $n$-dimensional ball around $0$ such that the following is true:
The $n$-dimensional ball is a ...
1
vote
1
answer
419
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
$u$ &...
- 121