Set In Rn Research Articles

A geometric characterization of a sharp Hardy inequality

In this paper, we prove that the distance function of an open connected set in Rn+1 with a C2 boundary is superharmonic in the distribution sense if and only if the boundary is weakly mean convex. We then prove that Hardy inequalities with a sharp constant hold on weakly mean convex C2 domains. Moreover, we show that the weakly mean convexity condition cannot be weakened. We also prove various improved Hardy inequalities on mean convex domains along the line of Brezis and Marcus (1997) [7].

On a space of smooth functions on a convex unbounded set in ℝn admitting holomorphic extension in ℂn

Abstract For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ℝn. Due to the conditions on M each function of this space admits a holomorphic extension in ℂn. In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions satisfying some weighted L 2-bounds in a domain of holomorphy in ℂn are obtained with the aid of L. Hörmander’s method of L 2-bounds for the $$\bar \partial$$ operator. Also, some new facts on the Fourier-Laplace transform of tempered distributions complementing some well-known results of V.S. Vladimirov are employed.

The space clcv(ℝ n ) with the Hausdorff-Bebutov metric and differential inclusions

The paper is devoted to studying the space of nonempty closed convex (but not necessarily compact) sets in ℝn, a dynamical system of translations, and existence theorems for differential inclusions. We make this space complete by equipping it with the Hausdorff-Bebutov metric. The investigation of these issues is important for certain problems of optimal control of asymptotic characteristics of a control system. For example, the problem \(\dot x = A(t,u)x\), (u, x) ∈ ℝm+n, λn(u(·))→ min, where λn(u(·)) is the largest Lyapunov exponent of the system {ie121-2} = A(t, u)x, leads to a differential inclusion with a noncompact right-hand side.

Large dimensional sets not containing a given angle

Abstract We say that a set in a Euclidean space does not contain an angle α if the angle determined by any three points of the set is not equal to α. The goal of this paper is to construct compact sets of large Hausdorff dimension that do not contain a given angle α ∈ (0,π). We will construct such sets in ℝn of Hausdorff dimension c(α)n with a positive c(α) depending only on α provided that α is different from π/3, π/2 and 2π/3. This improves on an earlier construction (due to several authors) that has dimension c(α) log n. The main result of the paper concerns the case of the angles π/3 and 2π/3. We present self-similar sets in ℝn of Hausdorff dimension $c{{\sqrt[3]{n}} \mathord{\left/ {\vphantom {{\sqrt[3]{n}} {\log n}}} \right. \kern-\nulldelimiterspace} {\log n}}$ with the property that they do not contain the angles π/3 and 2π/3. The constructed sets avoid not only the given angle α but also a small neighbourhood of α.

If you can hide behind it, can you hide inside it?

Let K and L be compact convex sets in Rn. Suppose that, for a given dimension 1 ≤ d ≤ n − 1, every d-dimensional orthogonal projection of L contains a translate of the corresponding projection of K. Does it follow that the original set L contains a translate of K? In other words, if K can be translated to “hide behind” L from any perspective, does it follow that K can “hide inside” L? A compact convex set L is defined to be d-decomposable if L is a direct Minkowski sum (affine Cartesian product) of two or more convex bodies each of dimension at most d. A compact convex set L is called d-reliable if, whenever each d-dimensional orthogonal projection of L contains a translate of the corresponding d-dimensional projection of K, it must follow that L contains a translate of K. It is shown that, for 1 ≤ d ≤ n− 1: (1) d-decomposability implies d-reliability. (2) A compact convex set L in Rn is d-reliable if and only if, for all m ≥ d+2, no m unit normals to regular boundary points of L form the outer unit normals of an (m− 1)-dimensional simplex. (3) Smooth convex bodies are not d-reliable. (4) A compact convex set L in Rn is 1-reliable if and only if L is 1-decomposable (i.e. a parallelotope). (5) A centrally symmetric compact convex set L in Rn is 2-reliable if and only if L is 2-decomposable. However, there are non-centered 2-reliable convex bodies that are not 2-decomposable. As a result of (5) above, the only reliable centrally symmetric covers in R3 from the perspective of 2-dimensional shadows are the affine convex cylinders (prisms). However, in dimensions greater than 3, it is shown that 3decomposability is only sufficient, and not necessary, for L to cover reliably with respect to 3-shadows, even when L is assumed to be centrally symmetric. Consider two compact convex subsetsK and L of n-dimensional Euclidean space. Suppose that, for a given dimension 1 ≤ d 0 is sufficiently small), but no longer fits inside D. The same construction works for any set K inscribed in D and having strictly less than unit diameter. Another Received by the editors June 4, 2009. 2000 Mathematics Subject Classification. Primary 52A20. c ©2011 Daniel A. Klain

On the Branch Points of Mappings with the Unbounded Coefficient of Quasiconformality

We study relations between the quantity characterizing the distortion of families of curves under a given mapping and the structure of the branch point set of this mapping. For n ⩽ 3 we establish that the image of the branch point set of an open discrete mapping with an isolated essential singularity is an unbounded set in Rn provided that the mapping satisfies certain geometric conditions controlling the distortion of concentric annuli centered at this point.

A sharp form of an embedding into multiple exponential spaces

Let Θ be a bounded open set in ℝ n , n ⩾ 2. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that $$ \sup \left\{ {\int_\Omega {\exp \left( {\left( {\frac{{\left| {f(x)} \right|}} {K}} \right)^{{n \mathord{\left/ {\vphantom {n {(n - 1)}}} \right. \kern-\nulldelimiterspace} {(n - 1)}}} } \right):f \in W_0^{1,n} (\Omega ),\left\| {\nabla f} \right\|_{L^n } \leqslant 1} } \right\} < \infty $$ . We extend this result to the situation in which the underlying space L n is replaced by the generalized Zygmund space L n logn−1 L log α log L (α < n − 1), the corresponding space of exponential growth then being given by a Young function which behaves like exp(exp(t n/(n−1−α))) for large t. We also discuss the case of an embedding into triple and other multiple exponential cases.

Maximal Lattice-Free Convex Sets in Linear Subspaces

We consider a model that arises in integer programming and show that all irredundant inequalities are obtained from maximal lattice-free convex sets in an affine subspace. We also show that these sets are polyhedra. The latter result extends a theorem of Lovász characterizing maximal lattice-free convex sets in ℝn.

Computation of the Metric Average of 2D Sets with Piecewise Linear Boundaries

The metric average is a binary operation between sets in Rn which is used in the approximation of set-valued functions. We introduce an algorithm that applies tools of computational geometry to the computation of the metric average of 2D sets with piecewise linear boundaries.

Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations

Forward, backward and elliptic Harnack inequalities for non-negative solutions of a class of singular, quasi-linear, parabolic equations, are established. These classes of singular equations include the p-Laplacean equation and equations of the porous medium type. Key novel points include form of a Harnack estimate backward in time, that has never been observed before, and measure theoretical proofs, as opposed to comparison principles. These Harnack estimates are established in the super-critical range (1.5) below. Such a range is optimal for a Harnack estimate to hold. Mathematics Subject Classification (2010): 35K65 (primary); 35B65, 35B45 (secondary). 1. Main results Let E be an open set in RN and for T > 0 let ET = E × (0, T ]. Let u be a weak solution u ∈ Cloc ( 0, T ; Lloc(E) ) ∩ L p loc(0, T ; W 1,p loc (E)) 1 < p < 2 (1.1) of a quasi-linear, singular parabolic equation of the type ut − div A(x, t, u, Du) = B(x, t, u, Du) weakly in ET (1.2) where the functions A : ET × RN+1 → RN and B : ET × RN+1 → R are only assumed to be measurable and subject to the structure conditions   A(x, t, u, Du) · Du ≥ Co|Du|p − C p |A(x, t, u, Du)| ≤ C1|Du|p−1 + C p−1 |B(x, t, u, Du)| ≤ C |Du|p−1 + C p a.e. in ET (1.3) This work has been partially supported by I.M.A.T.I. – C.N.R. – Pavia. DiBenedetto’s work partially supported by National Science Foundation grant DMS-0652385. Received March 9, 2009; accepted in revised form May 31, 2009.

Minimum Hausdorff distance under rigid motions and comparison of protein structures

Hausdorff distance between two compact sets, defined as the maximum distance from a point of one set to another set, has many application in computer science. It is a good measure for the similarity of two sets. This paper proves that the shape distance between two compact sets in Rn defined by minimum Hausdorff distance under rigid motions is a distance. The authors introduce similarity comparison problems in protein science, and propose that this measure may have good application to comparison of protein structure as well. For calculation of this distance, the authors give one dimensional formulas for problems (2, n), (3, 3), and (3, 4). These formulas can reduce time needed for solving these problems. The authors did some numerical experiments for (2, n). On these sets of data, this formula can reduce time needed to one fifteenth of the best algorithms known on average. As n increases, it would save more time.

Ned sets on a hyperplane

Under study are the sets in ℝn (NED sets) each of which does not affect the conformal capacity of any condenser with connected plates disjoint from this set. These sets are removable singularities of quasiconformal mappings, which explains our interest in them. For compact sets on a hyperplane we obtain a geometric criterion of the NED property; we point out a simple sufficient condition for an NED set in terms of the connected attainability of its points from its complement in the hyperplane. For compact sets on a hypersphere we obtain a criterion for an NED set in terms of the reduced module at a pair of points in its complement. We establish that a compact set on a hypersphere S, removable for the capacity in at least one spherical ring concentric with S and containing S, is an NED set.

The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field

Let R be a sufficiently saturated o-minimal expansion of a real closed field, let \( \mathcal{O} \) be the convex hull of ℚ in R, and let st: \( \mathcal{O}^n \) → ℝn be the standard part map. For X ⊆ Rn define st X:= st (X ∩ \( \mathcal{O}^n \)). We let ℝind be the structure with underlying set ℝ and expanded by all sets st X, where X ⊆ Rn is definable in R and n = 1, 2,.... We show that the subsets of ℝn that are definable in ℝind are exactly the finite unions of sets st X st Y, where X, Y ⊆ Rn are definable in R. A consequence of the proof is a partial answer to a question by Hrushovski, Peterzil and Pillay about the existence of measures with certain invariance properties on the lattice of bounded definable sets in Rn.

Tauberian conditions for geometric maximal operators

Let B \mathcal {B} be a collection of measurable sets in R n \mathbb {R}^{n} . The associated geometric maximal operator M B M_{\mathcal {B}} is defined on L 1 ( R n ) L^{1}(\mathbb {R}^n) by M B f ( x ) = sup x ∈ R ∈ B 1 | R | ∫ R | f | M_{\mathcal {B}}f(x) = \sup _{x \in R \in \mathcal {B}}\frac {1}{|R|}\int _{R}|f| . If α &gt; 0 \alpha &gt; 0 , M B M_\mathcal {B} is said to satisfy a Tauberian condition with respect to α \alpha if there exists a finite constant C C such that for all measurable sets E ⊂ R n E \subset \mathbb {R}^n the inequality | { x : M B χ E ( x ) &gt; α } | ≤ C | E | |\{x : M_{\mathcal {B}} \chi _{E}(x) &gt; \alpha \}| \leq C|E| holds. It is shown that if B \mathcal {B} is a homothecy invariant collection of convex sets in R n \mathbb {R}^{n} and the associated maximal operator M B M_{\mathcal {B}} satisfies a Tauberian condition with respect to some 0 &gt; α &gt; 1 0 &gt; \alpha &gt; 1 , then M B M_\mathcal {B} must satisfy a Tauberian condition with respect to γ \gamma for all γ &gt; 0 \gamma &gt; 0 and moreover M B M_{\mathcal {B}} is bounded on L p ( R n ) L^{p}(\mathbb {R}^{n}) for sufficiently large p p . As a corollary of these results it is shown that any density basis that is a homothecy invariant collection of convex sets in R n \mathbb {R}^{n} must differentiate L p ( R n ) L^{p}(\mathbb {R}^{n}) for sufficiently large 𝑝.

On the classification of solutions of -Δ𝑢=𝑒^{𝑢} on ℝ^{ℕ}: Stability outside a compact set and applications

In this short paper we prove that, for3≤N≤93 \le N \le 9, the problem−Δu=eu-\Delta u = e^uon the entire Euclidean spaceRN\mathbb {R}^Ndoes not admit any solution stable outside a compact set ofRN\mathbb {R}^N. This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems.

Variational Inequalities over Perturbed Polyhedral Convex Sets

This paper provides conditions for existence of a locally unique, Lipschitzian solution of a linear variational inequality posed over a polyhedral convex set in ℝn under perturbation of either or both of the constant term in the variational inequality and the right-hand side of the system of linear constraints defining its feasible set. Conditions for perturbation of just the constant term are well known. Here we show that a suitable extension of those conditions suffices for the more general case in which the right-hand side of the constraints varies also. As a consequence, we obtain existence, uniqueness, and Lipschitz continuity properties of solutions of nonlinear variational inequalities posed over perturbed polyhedral convex sets.

On sequence space representations of Hörmander–Beurling spaces

It is shown that B p ′ , 1 / k ˜ loc ( Ω ) is isomorphic to ( B p , k c ( Ω ) ) b ′ ( Ω open set in R n , 1 ⩽ p < ∞ , k Beurling–Björck weight) extending a Hörmander's result (the proof we give is valid in the vector-valued case, too). As a consequence, and using Vogt's representation theorems and weighted L p -spaces of entire analytic functions, a number of results on sequence space representations of Hörmander–Beurling are given.

A characterization of convex calibrable sets in [formula omitted] with respect to anisotropic norms

A set is called “calibrable” if its characteristic function is an eigenvector of the subgradient of the total variation. The main purpose of this paper is to characterize the “ϕ-calibrability” of bounded convex sets in RN with respect to a norm ϕ (called anisotropy in the sequel) by the anisotropic mean ϕ-curvature of its boundary. It extends to the anisotropic and crystalline cases the known analogous results in the Euclidean case. As a by-product of our analysis we prove that any convex body C satisfying a ϕ-ball condition contains a convex ϕ-calibrable set K such that, for any V∈[|K|,|C|], the subset of C of volume V which minimizes the ϕ-perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data the characteristic function of a bounded convex set.

Sufficient Conditions and Duality for a Class of Multiobjective Fractional Programming Problems with Higher-Order (F,α,ρ,d)-Convexity

In this paper, a class of multiobjective fractional programming problems (denoted by (MFP)) is considered. First, the concept of higher-order (F,α,ρ,d)-convexity of a function f:C→R with respect to the differentiable function φ:Rn×Rn→R is introduced, where C is an open convex set in Rn and α:C×C→R+∖{0} is a positive value function. And an important property, which the ratio of higher-order (F,α,ρ,d)-convex functions is also higher-order (F,α′,ρ′,d′)-convex, is proved. Under the higher-order (F,α,ρ,d)-convexity assumptions, an alternative theorem is also given. Then, some sufficient conditions characterizing properly (or weakly) efficient solutions of (MFP) are obtained from the above property and alternative theorem. Finally, a class of dual problems is formulated and appropriate duality theorems are proved.

Distributional dimension of fractal sets in local fields

The distributional dimension of fractal sets in ℝn has been systematically studied by Triebel by virtue of the theory of function spaces. In this paper, we first discuss some important properties about the B-type spaces and the F-type spaces on local fields, then we give the definition of the distributional dimension dimD in local fields and study the relations between distributional dimension and Hausdorff dimension. Moreover, the analysis expression of the Hausdorff dimension is given. Lastly, we define the Fourier dimension in local fields, and obtain the relations among all the three dimensions.