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1
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Obata’s Rigidity Theorem for Metric Measure Spaces

100%
Ketterer C.
Analysis and Geometry in Metric Spaces
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2015
|
tom 3
|
nr 1
EN
We prove Obata’s rigidity theorem for metric measure spaces that satisfy a Riemannian curvaturedimension condition. Additionally,we show that a lower bound K for the generalizedHessian of a sufficiently regular function u holds if and only if u is K-convex. A corollary is also a rigidity result for higher order eigenvalues.
2
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Are law-invariant risk functions concave on distributions?

100%
Acciaio B., Svindland G.
Dependence Modeling
|
2013
|
tom 1
54-64
EN
While it is reasonable to assume that convex combinations on the level of random variables lead to a reduction of risk (diversification effect), this is no more true on the level of distributions. In the latter case, taking convex combinations corresponds to adding a risk factor. Hence, whereas asking for convexity of risk functions defined on random variables makes sense, convexity is not a good property to require on risk functions defined on distributions. In this paper we study the interplay between convexity of law-invariant risk functions on random variables and convexity/concavity of their counterparts on distributions. We show that, given a law-invariant convex risk measure, on the level of distributions, if at all, concavity holds true. In particular, this is always the case under the additional assumption of comonotonicity.
3
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On relations among the generalized second-order directional derivatives

100%
Pastor K.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2001
|
tom 21
|
nr 2
235-247
EN
In the paper, we deal with the relations among several generalized second-order directional derivatives. The results partially solve the problem which of the second-order optimality conditions is more useful.
4
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On a problem of Matkowski

100%
Daróczy Z., Maksa G.
Colloquium Mathematicum
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1999
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tom 82
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nr 1
117-123
EN
We solve Matkowski's problem for strictly comparable quasi-arithmetic means.
5
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Applications of the ‘Ham Sandwich Theorem’ to Eigenvalues of the Laplacian

100%
Funano K.
Analysis and Geometry in Metric Spaces
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2016
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tom 4
|
nr 1
EN
We apply Gromov’s ham sandwich method to get: (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in Euclidean space.
6
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Information-type divergence when the likelihood ratios are bounded

100%
Rukhin A. L.
Applicationes Mathematicae
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1996-1997
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tom 24
|
nr 4
415-423
EN
The so-called ϕ-divergence is an important characteristic describing "dissimilarity" of two probability distributions. Many traditional measures of separation used in mathematical statistics and information theory, some of which are mentioned in the note, correspond to particular choices of this divergence. An upper bound on a ϕ-divergence between two probability distributions is derived when the likelihood ratio is bounded. The usefulness of this sharp bound is illustrated by several examples of familiar ϕ-divergences. An extension of this inequality to ϕ-divergences between a finite number of probability distributions with pairwise bounded likelihood ratios is also given.
7
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Convexity ranks in higher dimensions

88%
Kojman M.
Fundamenta Mathematicae
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2000
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tom 164
|
nr 2
143-163
EN
A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a Polish linear space. Then S is countably convex if and only if there exists $α < ω_1$ so that ϱ(x) < α for all x ∈ S. Classification of countably convex closed subsets of Polish linear spaces follows then easily. A similar classification (by a different rank function) was previously known for closed subset of $ℝ^2$ [3]. As an application of ϱ to Banach space geometry, it is proved that for every $α < ω_1$, the unit sphere of C(ωα) with the sup-norm has rank α. Furthermore, a countable compact metric space K is determined by the rank of the unit sphere of C(K) with the natural sup-norm: Theorem. If $K_1,K_1$ are countable compact metric spaces and $S_i$ is the unit sphere in $C(K_i)$ with the sup-norm, i = 1,2, then $ϱ(S_1) = ϱ(S_2)$ if and only if $K_1$ and $K_2$ are homeomorphic. Uncountably convex closed sets are also studied in dimension n > 2 and are seen to be drastically more complicated than uncountably convex closed subsets of $ℝ^2$
8
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Non compact boundaries of complex analytic varieties in Hilbert spaces

88%
Mongodi S., Saracco A.
Complex Manifolds
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2014
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tom 1
|
nr 1
EN
We treat the boundary problem for complex varieties with isolated singularities, of complex dimension greater than or equal to 3, non necessarily compact, which are contained in strongly convex, open subsets of a complex Hilbert space H.We deal with the problem by cutting with a family of complex hyperplanes and applying the already known result for the compact case.
9
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Median and quasi-median direct products of graphs

88%
Brešar B., Jha P., Klavžar S., Zmazek B.
Discussiones Mathematicae Graph Theory
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2005
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tom 25
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nr 1-2
183-196
EN
Median graphs are characterized among direct products of graphs on at least three vertices. Beside some trivial cases, it is shown that one component of G×P₃ is median if and only if G is a tree in that the distance between any two vertices of degree at least 3 is even. In addition, some partial results considering median graphs of the form G×K₂ are proved, and it is shown that the only nonbipartite quasi-median direct product is K₃×K₃.
10
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A Generalization of m-Convexity and a Sandwich Theorem

75%
Lara T., Matkowski J., Merentes N., Quintero R., Wróbel M.
Annales Mathematicae Silesianae
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2017
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tom 31
|
nr 1
107-126
EN
Functional inequalities generalizing m-convexity are considered. A result of a sandwich type is proved. Some applications are indicated.
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