Related papers: The projection onto the cross
In this note, we highlight some properties of the metric projection onto a closed convex in a Hilbert space. In particular, we use some recent results on fixed points of nonexpansive potential operators.
Packing problems, even of objects with regular geometries, are in general non-trivial. For few special shapes, the features of crystalline as well as random, irregular two-dimensional (2D) packings are known. The packing of 2D crosses does…
Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex…
We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear…
We provide formulas for projectors onto a polyhedral set, i.e. the intersection of a finite number of halfspaces. To this aim we formulate the problem of finding the projection as a convex optimization problem and we solve explicitly…
We generalize the natural cross ratio on the ideal boundary of a rank one symmetric spaces, or even $\mathrm{CAT}(-1)$ space, to higher rank symmetric spaces and (non-locally compact) Euclidean buildings - we obtain vector valued cross…
We fully characterize orthogonal projections of infinite right circular (round) cones in real Hilbert spaces. Another interpretation is that, given two vectors in a real Hilbert space, we establish the optimal estimate on the angle between…
Given an Euclidean space, this paper elucidates the topological link between the partial derivatives of the Minkowski functional associated to a set (assumed to be compact, convex, with a differentiable boundary and a non-empty interior)…
On R^n endowed with a riemannian metric of bounded nonpositive curvature, the weakly convex closed subsets are topologically trivial. The stability of such subsets under intersection characterizes the euclidean spaces.
The Euclidean projection onto a convex set is an important problem that arises in numerous constrained optimization tasks. Unfortunately, in many cases, computing projections is computationally demanding. In this work, we focus on…
We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr{H}$ generated by a sequence $\mathcal{V} = \{v_n\}_{n=0}^\infty$. The first main result of this paper provides a sufficient condition under which…
We provide a direct proof of a result regarding the asymptotic behavior of alternating nearest point projections onto two closed and convex sets in a Hilbert space. Our arguments are based on nonexpansive mapping theory.
We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the…
We provide a sufficient condition for a finite number of closed subspaces of a Hilbert space to be linearly independent and their sum to be closed. Under this condition a formula for the orthogonal projection onto the sum is given. We also…
A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
Let $\mathcal{H}$ be a (separable) Hilbert space and $\{e_k\}_{k\geq 1}$ a fixed orthonormal basis of $\mathcal{H}$. Motivated by many papers on scaled projections, angles of subspaces and oblique projections, we define and study the notion…
It is well-known that the sequence of iterations of the composition of projections onto closed affine subspaces converges linearly to the projection onto the intersection of the affine subspaces when the sum of the corresponding linear…
Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of…
In this paper, we tackle the long-standing challenges of ensemble control analysis and design using a convex-geometric approach in a Hilbert space setting. Specifically, we formulate the control of linear ensemble systems as a convex…