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Equidecomposability of Jordan domains under groups of isometries

M. Laczkovich (2003)

Fundamenta Mathematicae

Let G d denote the isometry group of ℝ d . We prove that if G is a paradoxical subgroup of G d then there exist G-equidecomposable Jordan domains with piecewise smooth boundaries and having different volumes. On the other hand, we construct a system ℱ d of Jordan domains with differentiable boundaries and of the same volume such that ℱ d has the cardinality of the continuum, and for every amenable subgroup G of G d , the elements of ℱ d are not G-equidecomposable; moreover, their interiors are not G-equidecomposable...

Equilibria in a class of games and topological results implying their existence.

R.S. Simon, S. Spiez, H. Torunczyk (2008)

RACSAM

We survey results related to the problem of the existence of equilibria in some classes of infinitely repeated two-person games of incomplete information on one side, first considered by Aumann, Maschler and Stearns. We generalize this setting to a broader one of principal-agent problems. We also discuss topological results needed, presenting them dually (using cohomology in place of homology) and more systematically than in our earlier papers.

Equivalence classes of Latin squares and nets in ℂ P 2

Corey Dunn, Matthew Miller, Max Wakefield, Sebastian Zwicknagl (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The fundamental combinatorial structure of a net in ℂ P 2 is its associated set of mutually orthogonal Latin squares. We define equivalence classes of sets of orthogonal Latin squares by label equivalences of the lines of the corresponding net in ℂ P 2 . Then we count these equivalence classes for small cases. Finally we prove that the realization spaces of these classes in ℂ P 2 are empty to show some non-existence results for 4-nets in ℂ P 2 .

Equivariant classification of 2-torus manifolds

Zhi Lü, Mikiya Masuda (2009)

Colloquium Mathematicae

We consider locally standard 2-torus manifolds, which are a generalization of small covers of Davis and Januszkiewicz and study their equivariant classification. We formulate a necessary and sufficient condition for two locally standard 2-torus manifolds over the same orbit space to be equivariantly homeomorphic. This leads us to count the equivariant homeomorphism classes of locally standard 2-torus manifolds with the same orbit space.

Essentially-Euclidean convex bodies

Alexander E. Litvak, Vitali D. Milman, Nicole Tomczak-Jaegermann (2010)

Studia Mathematica

In this note we introduce a notion of essentially-Euclidean normed spaces (and convex bodies). Roughly speaking, an n-dimensional space is λ-essentially-Euclidean (with 0 < λ < 1) if it has a [λn]-dimensional subspace which has further proportional-dimensional Euclidean subspaces of any proportion. We consider a space X₁ = (ℝⁿ,||·||₁) with the property that if a space X₂ = (ℝⁿ,||·||₂) is "not too far" from X₁ then there exists a [λn]-dimensional subspace E⊂ ℝⁿ such that E₁ = (E,||·||₁) and...

Estimates on inner and outer radii of unit balls in normed spaces

Horst Martini, Zokhrab Mustafaev (2011)

Colloquium Mathematicae

The purpose of this paper is to continue the investigations on extremal values for inner and outer radii of the unit ball of a finite-dimensional real Banach space for the Holmes-Thompson and Busemann measures. Furthermore, we give a related new characterization of ellipsoids in ℝ d via codimensional cross-section measures.

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