Related papers: On the Grassmann condition number

The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show…

We investigate the problem of finding inner ap-proximations of positive semidefinite (PSD) cones. We developa novel decomposition framework of the PSD cone by meansof conical combinations of smaller dimensional sub-cones. Weshow that many…

The Grassmann angle improves upon similar angles between subspaces that measure volume contraction in orthogonal projections. It works in real or complex spaces, with important differences, and is asymmetric, what makes it more efficient…

Affine Grassmannian has been favored for expressing proximity between lines and planes due to its theoretical exactness in measuring distances among features. Despite this advantage, the existing method can only measure the proximity…

We provide an affirmative answer to a variant of the Busemann-Petty problem, proposed by V.~Milman: Let $K$ be a convex body in ${\mathbb R}^n$ and let $D$ be a compact subset of ${\mathbb R}^n$ such that, for some $1\ls k\ls n-1$,…

A stability criterion is derived for self-similar solutions with perfect fluids which obey the equation of state $P=k\rho$ in general relativity. A wide class of self-similar solutions turn out to be unstable against the so-called kink…

The convex and metric structures underlying probabilistic physical theories are generally described in terms of base normed vector spaces. According to a recent proposal, the purely geometrical features of these spaces are appropriately…

Given a decomposition of a vector space $V=V_1\oplus V_2$, the direct product $\mathfrak{X}$ of the projective space $\mathbb{P}(V_1)$ with a Grassmann variety $\mathrm{Gr}_k(V)$ can be viewed as a double flag variety for the symmetric pair…

We study a class of design problems in solid mechanics, leading to a variation on the classical question of equi-dimensional embeddability of Riemannian manifolds. In this general new context, we derive a necessary and sufficient existence…

We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$, $m\leq n$, we define and prove the existence of a partition of…

In inverse problems, many conditional generative models approximate the posterior measure by minimizing a distance between the joint measure and its learned approximation. While this approach also controls the distance between the posterior…

We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility…

Given a convex body $K \subseteq \mathbb R^n$ in L\"owner position we study the problem of constructing a non-negative centered isotropic measure supported in the contact points, whose existence is guaranteed by John's Theorem. The method…

Let $ l =[l_0,l_1]$ be the directed line segment from $l_0\in {\mathbb R}^n$ to $l_1\in{\mathbb R}^n.$ Suppose $\bar l=[\bar l_0,\bar l_1]$ is a second segment of equal length such that $l, \bar l$ satisfy the "two sticks condition": $\|…

Solving the inverse kinematics problem is a fundamental challenge in motion planning, control, and calibration for articulated robots. Kinematic models for these robots are typically parametrized by joint angles, generating a complicated…

We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a…

We investigate the properties of certain elliptic systems leading, a~priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic Uhlenbeck structure, then the solution…

Fix an integer k, and let I(l), l=1,2,..., be a sequence of k-dimensional vectors of multiple Wiener-It\^o integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as l diverges,…

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where…

This paper studies approximate solutions of a linear fractional vector optimization problem without requiring boundedness of the constraint set. We establish necessary and sufficient conditions for approximating weakly efficient points of…