Displaying similar documents to “Convex integration with constraints and applications to phase transitions and partial differential equations”

The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology

Katsuro Sakai, Zhongqiang Yang (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let C o n v F ( ) be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that C o n v F ( ) × Q for every n > 1 whereas C o n v F ( ) × .

Operations between sets in geometry

Richard J. Gardner, Daniel Hug, Wolfgang Weil (2013)

Journal of the European Mathematical Society

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An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in n -dimensional Euclidean space n . It is proved that if n 2 , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, G L ( n ) covariant, and associative if and only if it is L p addition for some 1 p . It is also demonstrated...

Poincaré Inequalities and Moment Maps

Bo’az Klartag (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We discuss a method for obtaining Poincaré-type inequalities on arbitrary convex bodies in n . Our technique involves a dual version of Bochner’s formula and a certain moment map, and it also applies to some non-convex sets. In particular, we generalize the central limit theorem for convex bodies to a class of non-convex domains, including the unit balls of p -spaces in n for 0 < p < 1 .

On the ψ₂-behaviour of linear functionals on isotropic convex bodies

G. Paouris (2005)

Studia Mathematica

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The slicing problem can be reduced to the study of isotropic convex bodies K with d i a m ( K ) c n L K , where L K is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that | | · , θ | | ψ C L K for all θ in a subset U of S n - 1 with measure σ(U) ≥ 1 - exp(-c√n). However, there exist isotropic convex bodies K with uniformly bounded geometric distance from the Euclidean ball, such that m a x θ S n - 1 | | · , θ | | ψ c n L K . In a different direction, we show that good average ψ₂-behaviour of linear functionals...

Product property for capacities in N

Mirosław Baran, Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

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The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: C ν ( E × E ) = m i n ( C ν ( E ) , C ν ( E ) ) , where E j and ν j are respectively a compact set and a norm in N j (j = 1,2), and ν is a norm in N + N , ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of N , denote by C(E) the standard L-capacity and by ω E the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

The Young inequality and the Δ₂-condition

Philippe Laurençot (2002)

Colloquium Mathematicae

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If φ: [0,∞) → [0,∞) is a convex function with φ(0) = 0 and conjugate function φ*, the inequality x y ε φ ( x ) + C ε φ * ( y ) is shown to hold true for every ε ∈ (0,∞) if and only if φ* satisfies the Δ₂-condition.

Minimal multi-convex projections

Grzegorz Lewicki, Michael Prophet (2007)

Studia Mathematica

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We say that a function from X = C L [ 0 , 1 ] is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Πₙ ⊂ X, where Πₙ denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general “shape”...

Quantitative stability for sumsets in n

Alessio Figalli, David Jerison (2015)

Journal of the European Mathematical Society

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Given a measurable set A n of positive measure, it is not difficult to show that | A + A | = | 2 A | if and only if A is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If ( | A + A | - | 2 A | ) / | A | is small, is A close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between A and its convex hull in terms of ( | A + A | - | 2 A | ) / | A | .

A note on the commutator of two operators on a locally convex space

Edvard Kramar (2016)

Commentationes Mathematicae Universitatis Carolinae

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Denote by C the commutator A B - B A of two bounded operators A and B acting on a locally convex topological vector space. If A C - C A = 0 , we show that C is a quasinilpotent operator and we prove that if A C - C A is a compact operator, then C is a Riesz operator.

Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

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We consider the class 0 of sense-preserving harmonic functions f = h + g ¯ defined in the unit disk | z | < 1 and normalized so that h ( 0 ) = 0 = h ' ( 0 ) - 1 and g ( 0 ) = 0 = g ' ( 0 ) , where h and g are analytic in the unit disk. In the first part of the article we present two classes 𝒫 H 0 ( α ) and 𝒢 H 0 ( β ) of functions from 0 and show that if f 𝒫 H 0 ( α ) and F 𝒢 H 0 ( β ) , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections...

On the Schröder equation

M. Kuczma

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CONTENTSPART IIntroduction............................................................................................... 31. General solution.................................................................................. 42. Preliminaries and notation................................................................ 53. C p solutions in *................................................ 74. Change of variables..............................................................................

Matrix subspaces of L₁

Gideon Schechtman (2013)

Studia Mathematica

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If E = e i and F = f i are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices a i , j with norm | | a i , j | | E ( F ) = | | k | | l a k , l f l | | e k | | embeds into L₁. This generalizes a recent result of Prochno and Schütt.

Geometry and inequalities of geometric mean

Trung Hoa Dinh, Sima Ahsani, Tin-Yau Tam (2016)

Czechoslovak Mathematical Journal

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We study some geometric properties associated with the t -geometric means A t B : = A 1 / 2 ( A - 1 / 2 B A - 1 / 2 ) t A 1 / 2 of two n × n positive definite matrices A and B . Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding...

A symmetry problem in the calculus of variations

Graziano Crasta (2006)

Journal of the European Mathematical Society

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We consider the integral functional J ( u ) = Ω [ f ( | D u | ) u ] d x , u W 0 1 , 1 ( Ω ) , where Ω n , n 2 , is a nonempty bounded connected open subset of n with smooth boundary, and s f ( | s | ) is a convex, differentiable function. We prove that if J admits a minimizer in W 0 1 , 1 ( Ω ) depending only on the distance from the boundary of Ω , then Ω must be a ball.

Łojasiewicz-Siciak condition for the pluricomplex Green function

Marta Kosek (2011)

Banach Center Publications

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A compact set K N satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that V K ( z ) B ( d i s t ( z , K ) ) β if dist(z,K) ≤ 1. (LS) Here V K denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).

Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment

Agnieszka Bogdewicz, Jerzy Grzybowski (2009)

Banach Center Publications

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Let ( , | | · | | ) be a Minkowski space with a unit ball and let ϱ H be the Hausdorff metric induced by | | · | | in the hyperspace of convex bodies (nonempty, compact, convex subsets of ℝ). R. Schneider [RSP] characterized pairs of elements of which can be joined by unique metric segments with respect to ϱ H B for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace ( ² , ϱ H ) over any two-dimensional Minkowski space.

The vertex detour hull number of a graph

A.P. Santhakumaran, S.V. Ullas Chandran (2012)

Discussiones Mathematicae Graph Theory

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For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), I D [ S ] = x , y S I D [ x , y ] . A set S of vertices is a detour convex set if I D [ S ] = S . The detour convex hull [ S ] D is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of...

The joint essential numerical range of operators: convexity and related results

Chi-Kwong Li, Yiu-Tung Poon (2009)

Studia Mathematica

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Let W(A) and W e ( A ) be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that W e ( A ) is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, W e ( A ) can be obtained as the intersection of all sets of the form c l ( W ( A , . . . , A i + 1 , A i + F , A i + 1 , . . . , A ) ) , where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in...