Contractibility of the hyperspace of meager subcontinua, two examples

Edgeworth's conjecture in economies with a continuum of agents and commodities

We show that each infinite-dimensional reflexive Banach space (X,left| cdot right| _X) has an equivalent norm left| cdot right| _{X,0} such that (X,left| cdot right| _{X,0}) is LUR and contains a diametrically complete set with empty interior. We also prove that after a suitable equivalent renorming, the Banach space C([0,1],{mathbb {R}}) contains a constant width set with empty interior.

We study the size of the sets of gradients of bump functions on the Hilbert space ℓ 2 , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in ℓ 2 can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space ℓ 2 can be uniformly approximated by C 1 smooth Lipschitz functions ψ so that the cones generated by the ranges of its derivatives ψ ' (ℓ 2 ) have empty interior. This implies that there are C 1 smooth Lipschitz bumps in ℓ 2 so that the cones generated by their sets of gradients have empty interior. On the other hand, we construct C 1 -smooth bounded starlike bodies A⊂ℓ 2 , which approximate the unit ball, so that the cones generated by the hyperplanes which are tangent to A have empty interior as well. We also explain why this is the best answer to the above questions that one can expect.

Our first result says that every real or complex infinite-dimensional normed space has an unbounded absolutely convex and absorbing subset with empty interior. As a consequence, a real normed space is finite-dimensional if and only if every convex subset containing 0 whose linear span is the whole space has non-empty interior. In our second result we prove that every real or complex separable normed space with dimension greater than 1 contains a balanced and absorbing subset with empty interior which is dense in the unit ball. Explicit constructions of these subsets are given.

In [Problems on self-similar sets and self-affine sets: An update, Birkhäuser, Basel, 2000] Peres and Solomyak ask the question: Do there exist self-similar sets with positive Lebesgue measure and empty interior? This question was answered in the affirmative by Csörnyei et al. in 2006. The authors of that paper gave a parameterised family of iterated function systems for which almost all of the corresponding self-similar sets satisfied the required properties. They did not however provide an explicit example. Motivated by a desire to construct an explicit example, we provide an explicit construction of an infinitely generated self-similar set with positive Lebesgue measure and empty interior.

For 0<ϵ≤1 and an element a of a complex unital Banach algebra A, we prove the following two topological properties about the level sets of the condition spectrum. (1) If ϵ=1, then the 1-level set of the condition spectrum of a has an empty interior unless a is a scalar multiple of the unity. (2) If 0<ϵ<1, then the ϵ-level set of the condition spectrum of a has an empty interior in the unbounded component of the resolvent set of a. Further, we show that, if the Banach space X is complex uniformly convex or if X∗ is complex uniformly convex, then, for any operator T acting on X, the level set of the ϵ-condition spectrum of T has an empty interior.

An extension of the Sard–Smale Theorem to convex domains with an empty interior

Existence of diametrically complete sets with empty interior in reflexive and separable Banach spaces

Here we show existence of numerous subsets of Euclidean and metric spaces\nthat, despite having empty interior, still support Poincar\\'e inequalities.\nMost importantly, our methods do not depend on any rectilinear or self-similar\nstructure of the underlying space. We instead employ the notion of uniform\ndomain of Martio and Sarvas. Our condition relies on the measure density of\nsuch subsets, as well as the regularity and relative separation of their\nboundary components.\n In doing so, our results hold true for metric spaces equipped with doubling\nmeasures and Poincar\\'e inequalities in general, and for the Heisenberg groups\nin particular. To our knowledge, these are the first examples of such subsets\non any step-2 Carnot group. Such subsets also give, in general, new examples of\nSobolev extension domains on doubling metric measure spaces. When specialized\nto the plane, we give general sufficient conditions for planar subsets,\npossibly with empty interior, to be Ahlfors 2-regular and to satisfy a\n(1,2)-Poincar\\'e inequality.\n In the Euclidean case, our construction also covers the non-self-similar\nSierpi\\'nski carpets of Mackay, Tyson, and Wildrick, as well as higher\ndimensional analogues not treated in the literature. The analysis of the\nPoincar\\'e inequality with exponent p=1, for these carpets and their higher\ndimensional analogues, includes a new way of proving an isoperimetric\ninequality on a space without constructing Semmes families of curves.\n

The problem of a spherically symmetric charged thin shell of dust collapsing gravitationally into a charged Reissner–Nordström black hole in d space–time dimensions is studied within the theory of general relativity. Static charged shells in such a background are also analyzed. First, a derivation of the equation of motion of such a shell in a d-dimensional space–time is given. Then, a proof of the cosmic censorship conjecture in a charged collapsing framework is presented, and a useful constraint which leads to an upper bound for the rest mass of a charged shell with an empty interior is derived. It is also proved that a shell with total mass equal to charge, i.e. an extremal shell, in an empty interior, can only stay in neutral equilibrium outside its gravitational radius. This implies that it is not possible to generate a regular extremal black hole by placing an extremal dust thin shell within its own gravitational radius. Moreover, it is shown, for an empty interior, that the rest mass of the shell is limited from above. Then, several types of behavior of oscillatory charged shells are studied. In the presence of a horizon, it is shown that an oscillatory shell always enters the horizon and reemerges in a new asymptotically flat region of the extended Reissner–Nordström space–time. On the other hand, for an overcharged interior, i.e. a shell with no horizons, an example showing that the shell can achieve a stable equilibrium position is presented. The results presented have applications in brane scenarios with extra large dimensions, where the creation of tiny higher-dimensional charged black holes in current particle accelerators might be a real possibility, and generalize to higher dimensions previous calculations on the dynamics of charged shells in four dimensions.

As a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space X is very smooth or its bidual satisfies the w*-Mazur intersection property, then either X is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if X has the Mazur intersection property and so, if the norm of X is Fréchet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion.

We prove that for any K-analytic subsets A,B of a locally compact group X if the product AB has empty interior (and is meager) in X, then one of the sets A or B can be covered by countably many closed nowhere dense subsets (of Haar measure zero) in X. This implies that a K-analytic subset A of X can be covered by countably many closed Haar-null sets if the set AAAA has an empty interior in X. It also implies that every non-open K-analytic subgroup of a locally compact group X can be covered by countably many closed Haar-null sets in X (for analytic subgroups of the real line this fact was proved by Laczkovich in 1998). Applying this result to the Kuczma–Ger classes, we prove that an additive function f:X→R on a locally compact topological group X is continuous if and only if f is upper bounded on some K-analytic subset A⊆X that cannot be covered by countably many closed Haar-null sets.

We prove that for each nonseparable and reflexive Banach space $(X,\|\cdot\|_X)$ with the nonstrict Opial and Kadec–Klee properties, there exists an equivalent norm $\|\cdot\|_0$ such that the Banach space $(X,\|\cdot\|_0)$ is LUR and contains a diametrically complete set with empty interior.

In the context of the Franks–Misiurewicz conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterize these maps by the existence of an invariant ‘foliation’ by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalizing a well-known result of Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector. As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior.

In this paper, we first establish chain rules and sum rules for variational sets of type 2. For their applications, optimality conditions of two particular optimization problems are discussed. Then, we obtain higher-order optimality conditions for proper Henig solutions of a set-valued optimization problem in terms of variational sets of type 2 when ordering cones have empty interior.