Nonempty Interior Research Articles

Quantifying Minimal Noncollinearity among Random Points

Let $\varphi_ {n, K} $ denote the largest angle in all the triangles with vertices among the $ n $ points selected at random in a compact convex subset $ K $ of $\mathbb {R}^ d $ with nonempty interior, where $ d\ge2 $. It is shown that the distribution of the random variable $\lambda_d (K)\,\frac {n^ 3}{3!}\,(\pi-\varphi_ {n, K})^{d-1} $, where $\lambda_d (K) $ is a certain positive real number which depends only on the dimension $d$ and the shape of $K$, converges to the standard exponential distribution as $n\to\infty$. By using the Steiner symmetrization, it is also shown that $\lambda_d (K)$, which is referred to in the paper as the elongation of $K$, attains its minimum if and only if $K$ is a ball $B^{(d)}$ in $\mathbf {R}^d$. Finally, the asymptotics of $\lambda_d(B^{(d)})$ for large $d$ is determined.

Nonlinear Conjugate Gradient Methods for Vector Optimization

In this work, we propose nonlinear conjugate gradient methods for finding critical points of vector-valued functions with respect to the partial order induced by a closed, convex, and pointed cone with nonempty interior. No convexity assumption is made on the objectives. The concepts of Wolfe and Zoutendjik conditions are extended for the vector-valued optimization. In particular, we show that there exist intervals of step sizes satisfying the Wolfe-type conditions. The convergence analysis covers the vector extensions of the Fletcher--Reeves, conjugate descent, Dai--Yuan, Polak--Ribière--Polyak, and Hestenes--Stiefel parameters that retrieve the classical ones in the scalar minimization case. Under inexact line searches and without regular restarts, we prove that the sequences generated by the proposed methods find points that satisfy the first-order necessary condition for Pareto-optimality. Numerical experiments illustrating the practical behavior of the methods are presented.

On the Exactness of Lasserre Relaxations for Compact Convex Basic Closed Semialgebraic Sets

Consider a finite system of nonstrict real polynomial inequalities and suppose its solution set $S\subseteq\mathbb R^n$ is convex, has nonempty interior, and is compact. Suppose that the system sat...

Spatially independent martingales, intersections, and applications

We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrised measures $\{\eta_t\}_t$, and show that under some natural checkable conditions, a.s. the total measure of the intersections is Holder continuous as a function of $t$. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. Łaba in connection to the restriction problem for fractal measures.

Discovering the interior of black holes

The detection of gravitational waves (GWs) from black hole (BH) mergers provides an inroad toward probing the interior of astrophysical BHs. The general-relativistic description of the BH interior is that of empty spacetime with a (possibly) singular core. Recently, however, the hypothesis that the BH interior does not exist has been gaining traction, as it provides a means for resolving the BH information-loss problem. Here, we propose a simple method for answering the following question: Does the BH interior exist and, if so, does it contain some distribution of matter or is it mostly empty? Our proposal is premised on the idea that, similar to the case of relativistic, ultra-compact stars, any BH-like object whose interior has some matter distribution should support fluid modes in addition to the conventional spacetime modes. In particular, the Coriolis-induced Rossby (r-) modes, whose spectrum is mostly insensitive to the composition of the interior matter, should be a universal feature of such BH-like objects. In fact, the frequency and damping time of these modes are determined by only the object's mass and speed of rotation. The r-modes oscillate at a lower frequency, decay at a slower rate and produce weaker GWs than do modes of the spacetime class. Hence, they imprint a model-insensitive signature of a non-empty interior in the GW spectrum resulting from a BH merger. We find that future GW detectors, such as Advanced LIGO with its design sensitivity, have the potential of detecting such r-modes if the amount of GWs leaking out quantum mechanically from the interior of a BH-like object is sufficiently large.

Inscribed Balls and Their Centers

A ball of maximal radius inscribed in a convex closed bounded set with a nonempty interior is considered in the class of uniformly convex Banach spaces. It is shown that, under certain conditions, the centers of inscribed balls form a uniformly continuous (as a set function) set-valued mapping in the Hausdorff metric. In a finite-dimensional space of dimension n, the set of centers of balls inscribed in polyhedra with a fixed collection of normals satisfies the Lipschitz condition with respect to sets in the Hausdorff metric. A Lipschitz continuous single-valued selector of the set of centers of balls inscribed in such polyhedra can be found by solving n + 1 linear programming problems.

Arithmetic patches, weak tangents, and dimension

We investigate the relationships between several classical notions in arithmetic combinatorics and geometry including: the presence (or lack of) arithmetic progressions (or patches in dimensions $\geq 2$); the structure of tangent sets; and the Assouad dimension. We begin by extending a recent result of Dyatlov and Zahl by showing that a set cannot contain arbitrarily large arithmetic progressions (patches) if it has Assouad dimension strictly smaller than the ambient spatial dimension. Seeking a partial converse, we go on to prove that having Assouad dimension equal to the ambient spatial dimension is equivalent to having weak tangents with non-empty interior and to `asymptotically' containing arbitrarily large arithmetic patches. We present some applications of our results concerning sets of integers, which include a weak solution to the Erd\"os-Tur\'an conjecture on arithmetic progressions.

When is multiplication in a Banach algebra open?

We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in ℓ1(N0) is open. By appealing to ultraproduct techniques, we demonstrate that neither in ℓ1(Z) nor in ℓ1(Q) convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed.

Hamiltonian C0-continuity of Lagrangian capacity on the cotangent bundle

Partially motivated by the study of topological Hamiltonian dynamics, we prove the following C0-continuity of the Lagrangian capacity function γlag: γlag(ϕH1(oN)):=ρlag(H;1)−ρlag(H;[pt]#)→0, as ϕH1→id, provided the H’s satisfy suppXH⊂DR(T∗N)∖oB for some R>0 and a closed subset B⊂N with nonempty interior. We also provide an estimate of the capacity in terms of the C0-distance of dC0(ϕH1,id) and the subset B⊂N relative to T∗N.

Entropy of Convex Functions on ℝ d.

Let Ω be a bounded closed convex set in ℝ d with non-empty interior, and let 𝒞 r (Ω) be the class of convex functions on Ω with Lr -norm bounded by 1. We obtain sharp estimates of the ε-entropy of 𝒞 r (Ω) under Lp (Ω) metrics, 1 ≤ p < r ≤ ∞. In particular, the results imply that the universal lower bound ε-d/2 is also an upper bound for all d-polytopes, and the universal upper bound of [Formula: see text] for [Formula: see text] is attained by the closed unit ball. While a general convex body can be approximated by inscribed polytopes, the entropy rate does not carry over to the limiting body. Our results have applications to questions concerning rates of convergence of nonparametric estimators of high-dimensional shape-constrained functions.

The general Pettis–Sugeno integral of vector multifunctions relative to a vector fuzzy multimeasure

In this paper, we define and study the general Pettis–Sugeno integral of a vector multifunction relative to a vector fuzzy multimeasure, both taking values in a locally convex space X, ordered by a closed convex pointed cone X+, with nonempty interior. Several classic properties of this integral and some estimation and comparative results with the Pettis–Lebesgue integral are established.

Error diffusion on acute simplices: invariant tiles

We study the absorbing invariant set of a dynamical system defined by a map derived from Error Diffusion, a greedy online approximation algorithm that minimizes the (Euclidean) norm of the cumulated error. This algorithm assigns a sequence of outputs, each a vertex of some polytope, to any sequence of inputs in that polytope. Here, the polytope is assumed to be a simplex that is acute, meaning that the pairs of distinct external normal vectors to the codimension-one faces form obtuse angles. The input is assumed to be constant. The map is a system which consists of piecewise translations acting on the partition of an affine space into the Vorono¨i regions defined (once tie-breaking is resolved) by the vertices of the polytope. The translation vector in each partition piece is the difference between the input modified by adding the cumulated error and the corresponding vertex. When the polytope is a simplex such piecewise translation projects onto a translation of a torus. We prove that if the projected translation is ergodic, then there is an invariant absorbing set of the piecewise translation which is a fundamental set for the lattice generated by the simplex and which is a limit set of any bounded set with nonempty interior.

Constrained controllability of fractional linear systems with delays in control

The paper presents finite-dimensional dynamical control systems described by linear fractional-order state equations with multiple delays in control. The constrained controls are considered. The set of admissible control values is assumed to be a compact set containing 0, a convex and compact set containing 0 in its interior, a cone with vertex at zero and a nonempty interior, or a convex and close cone with nonempty interior and vertex at zero. The definitions of various types of constrained controllability of the linear fractional systems with delays in control are formulated. New necessary and sufficient conditions for constrained relative controllability of fractional-order control systems with delayed controls are established and proved. Numerical examples are provided to illustrate the obtained theoretical results.

Diffeomorphism groups of compact convex sets

For K⊆Rn a compact convex subset with non-empty interior, let Diff∂K(K) be the group of all C∞-diffeomorphisms of K which fix ∂K pointwise. We show that Diff∂K(K) is a C0-regular infinite-dimensional Lie group. As a byproduct, we obtain results concerning solutions to ordinary differential equations on compact convex sets.

Bifurcations in the elementary Desboves family

We give an example of a family of endomorphisms of $\mathbb{P}^2(\mathbb{C})$ whose Julia set depends continuously on the parameter and whose bifurcation locus has non-empty interior.

Characterizations of set order relations and constrained set optimization problems via oriented distance function

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function.

On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect

We study a family of double confluent Heun equations that are lin-earizations of nonlinear equations on two-torus modeling the Joseph-son effect in superconductivity. They have the form LE = 0, where L = L λ,µ,n is a family of differential operators of order two acting on germs of holomorphic functions in one complex variable. They depend on parameters λ, n ∈ R, µ > 0, λ + µ 2 ≡ const > 0. We show that for every b ∈ C and n ∈ R satisfying a certain non-resonance condition and every parameter values λ, µ there exists a unique entire function f ± : C → C (up to multiplicative constant) such that z −b L(z b f ± (z ±1)) = d 0± + d 1± z for some d 0± , d 1± ∈ C. The latter d j,± are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those parameter values for which the monodromy operator of the corresponding Heun equation has given eigenvalues. This yields the description of the non-integer level curves of the rotation number of the family of equations on two-torus as a function of parameters. In the particular case, when the monodromy is parabolic (has multiple eigenvalue), we get the complete description of those parameter values that correspond to the boundaries of the phase-lock areas: integer level sets of the rotation number, which have non-empty interiors.

Tilings of Normed Spaces

Abstract By a tiling of a topological linear space X, we mean a covering of X by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite dimensional spaceswas initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study the existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space X, our main results are the following. (i) X admits no tiling by Fréchet smooth bounded tiles. (ii) If X is locally uniformly rotund (LUR), it does not admit any tiling by balls. (iii) On the other hand, some spaces, г uncountable, do admit a tiling by pairwise disjoint LUR bounded tiles.

Source conditions and accuracy estimates in Tikhonov’s scheme of solving ill-posed nonconvex optimization problems

Abstract We obtain rate of convergence estimates for approximations delivered by Tikhonov’s scheme of solving ill-posed nonconvex optimization problems in a Hilbert space. The problems under investigation involve minimization of twice Frechet differentiable functionals given with errors on a closed convex set having a nonempty interior and smooth boundary. Assuming that the desired solution satisfies an appropriate source condition which includes the second derivative of the cost functional and depends on the geometry of constraints near the solution, we establish accuracy estimates in terms of the error level. Both the a priori and a posteriori parameter choice rules are analyzed.

Critical solutions of nonlinear equations: local attraction for Newton-type methods

We show that if the equation mapping is 2-regular at a solution in some nonzero direction in the null space of its Jacobian (in which case this solution is critical; in particular, the local Lipschitzian error bound does not hold), then this direction defines a star-like domain with nonempty interior from which the iterates generated by a certain class of Newton-type methods necessarily converge to the solution in question. This is despite the solution being degenerate, and possibly non-isolated (so that there are other solutions nearby). In this sense, Newtonian iterates are attracted to the specific (critical) solution. Those results are related to the ones due to A. Griewank for the basic Newton method but are also applicable, for example, to some methods developed specially for tackling the case of potentially non-isolated solutions, including the Levenberg–Marquardt and the LP-Newton methods for equations, and the stabilized sequential quadratic programming for optimization.