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On closed sets with convex projections in Hilbert space

Stoyu Barov, Jan J. Dijkstra (2007)

Fundamenta Mathematicae

Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set k ( B ) consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that k ( B ) is precisely the intersection of all k-imitations...

On cyclic α(·)-monotone multifunctions

S. Rolewicz (2000)

Studia Mathematica

Let (X,d) be a metric space. Let Φ be a linear family of real-valued functions defined on X. Let Γ : X → 2 Φ be a maximal cyclic α(·)-monotone multifunction with non-empty values. We give a sufficient condition on α(·) and Φ for the following generalization of the Rockafellar theorem to hold. There is a function f on X, weakly Φ-convex with modulus α(·), such that Γ is the weak Φ-subdifferential of f with modulus α(·), Γ ( x ) = ∂ Φ - α f | x .

On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space

Stoyu Barov, Jan J. Dijkstra (2016)

Fundamenta Mathematicae

Let be a Euclidean space or the Hilbert space ℓ², let k ∈ ℕ with k < dim , and let B be convex and closed in . Let be a collection of linear k-subspaces of . A set C ⊂ is called a -imitation of B if B and C have identical orthogonal projections along every P ∈ . An extremal point of B with respect to the projections under is a point that all closed subsets of B that are -imitations of B have in common. A point x of B is called exposed by if there is a P ∈ such that (x+P) ∩ B = x. In the present...

On Gaussian Brunn-Minkowski inequalities

Franck Barthe, Nolwen Huet (2009)

Studia Mathematica

We are interested in Gaussian versions of the classical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrhard inequality for m Borel or convex sets based on a previous work by Borell. Our method also yields semigroup proofs of the geometric Brascamp-Lieb inequality and of its reverse form, which follow exactly the same lines.

On generalizations of fuzzy metric spaces

Yi Shi, Wei Yao (2023)

Kybernetika

The aim of the paper is to present three-variable generalizations of fuzzy metric spaces in sense of George and Veeramani from functional and topological points of view, respectively. From the viewpoint of functional generalization, we introduce a notion of generalized fuzzy 2-metric spaces, study their topological properties, and point out that it is also a common generalization of both tripled fuzzy metric spaces proposed by Tian et al. and ℳ -fuzzy metric spaces proposed by Sedghi and Shobe. Since...

On Gnomons

Jan M. Aarts, Robbert. J. Fokkink (2003)

Matematički Vesnik

On Hadwiger's problem on inner parallel bodies

Eugenia Saorín (2009)

Banach Center Publications

We consider the problem of classifying the convex bodies in the 3-dimensional space depending on the differentiability of their associated quermassintegrals with respect to the one-parameter-depending family given by the inner/outer parallel bodies. It turns out that this problem is closely related to some behavior of the roots of the 3-dimensional Steiner polynomial.

On hyperbolic virtual polytopes and hyperbolic fans

Gaiane Panina (2006)

Open Mathematics

Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ ℝ3 be a smooth convex body. If for a constant C, at every point of ∂K, we have R 1 ≤ C ≤ R 2 then K is a ball. (R 1 and R 2 stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic...

On hyperplanes and semispaces in max–min convex geometry

Viorel Nitica, Sergeĭ Sergeev (2010)

Kybernetika

The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry.

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