Related papers: Affine Subspace Concentration Conditions for Cente…
This paper is devoted to the complete classification of space curves under affine transformations in the view of Cartan's theorem. Spivak has introduced the method but has not found the invariants. Furthermore, for the first time, we…
This note provides a simple proof for the equality between the normalized volume of a convex polytope with $m$ vertices and the mixed volume of $m$ simplices and thus shows the seemingly restrictive problem of computing mixed volume of…
In this paper, we study main properties of cone normed spaces, and prove some theorems of weighted means in cone normed spaces.
We identify new sufficiency conditions for coercivity of general multivariate polynomials $f\in\mathbb{R}[x]$ which are expressed in terms of their Newton polytopes at infinity and which consist of a system of affine-linear inequalities in…
This note is purely expository. A subset N of the plane is affine ambient homogeneous if for each x,y in N there exists an affine transformation taking x to y and N to itself. The result of D. Repovs, E. V. Scepin and the author on such…
Our main contribution is a concentration inequality for the symmetric volume difference of a $ C^2 $ convex body with positive Gaussian curvature and a circumscribed random polytope with a restricted number of facets, for any probability…
The polytope containment problem is deciding whether a polytope is a contained within another polytope. This problem is rooted in computational convexity, and arises in applications such as verification and control of dynamical systems. The…
We find minimal and maximal length of intersections of lines at a fixed distance to the origin with the cross-polytope. We also find maximal volume noncentral sections of the cross-polypote by hyperplanes which are at a fixed large distance…
Frank-Wolfe methods are popular for optimization over a polytope. One of the reasons is because they do not need projection onto the polytope but only linear optimization over it. To understand its complexity, Lacoste-Julien and Jaggi…
We study minimizing cones in the Alt-Phillips problem when the exponent {\gamma} is close to 1. When {\gamma} converges to 1, we show that the cones concentrate around symmetric solutions to the classical obstacle problem. To be precise,…
Mirkovic-Vilonen (MV) polytopes have proven to be a useful tool in understanding and unifying many constructions of crystals for finite-type Kac-Moody algebras. These polytopes arise naturally in many places, including the affine…
This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another…
The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic…
We give necessary and sufficient conditions for an affine deformation of a Schottky subgroup of O(2,1) to act properly on affine space. There exists a real-valued biaffine map between the cohomology of the Schottky group and the space of…
Among all affine, flat, finitely presented group schemes, we focus on those that are pure, this includes all groups which are extensions of a finite locally free group by a group with connected fibres. We prove that over an arbitrary base…
In (the surface of) a convex polytope P^n in R^n+1, for small prescribed volume, geodesic balls about some vertex minimize perimeter. This revision corrects a mistake in the mass bound argument in the proof of Theorem 3.8.
In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside a closed disc centered at the origin in the complex plane and deduce the irreducibility over the ring of…
Volumes of moduli spaces of hyperbolic cone surfaces were previously defined and computed when the angles of the cone singularities are at most 2pi. We propose a general definition of these volumes without restriction on the angles. This…
We propose halfspace depth concepts for scatter, concentration and shape matrices. For scatter matrices, our concept is similar to those from Chen, Gao and Ren (2017) and Zhang (2002). Rather than focusing, as in these earlier works, on…
We classify all tuples of lattice polyhedra of relative mixed volume 1 and all minimal (by inclusion) tuples of polyhedra of relative mixed volume 2. We also prove a conjecture by A. Esterov, which states that all tuples with finite…