Circumradius versus side lengths of triangles in linear normed spaces

Gennadiy Averkov

Displaying similar documents to “Circumradius versus side lengths of triangles in linear normed spaces”

On area and side lengths of triangles in normed planes

Gennadiy Averkov, Horst Martini (2009)

Colloquium Mathematicae

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Let $ℳ^{d}$ be a d-dimensional normed space with norm ||·|| and let B be the unit ball in $ℳ^{d}$ . Let us fix a Lebesgue measure $V_{B}$ in $ℳ^{d}$ with $V_{B} (B) = 1$ . This measure will play the role of the volume in $ℳ^{d}$ . We consider an arbitrary simplex T in $ℳ^{d}$ with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of $V_{B} (T)$ are determined. For d ≥ 3 it is noticed that the tight lower bound of $V_{B} (T)$ is zero.

$L_{1}$ contains every two-dimensional normed space

David Yost (1988)

Annales Polonici Mathematici

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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 $d i a m (K) \leq c \sqrt n L_{K}$ , where $L_{K}$ is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that ${| | ⟨ \cdot, θ ⟩ | |}_{ψ ₂} \leq C L_{K}$ 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 $m a x_{θ \in S^{n - 1}} {| | ⟨ \cdot, θ ⟩ | |}_{ψ ₂} \geq c ∜ n L_{K}$ . In a different direction, we show that good average ψ₂-behaviour of linear functionals...

A theorem on isotropic spaces

Félix Cabello Sánchez (1999)

Studia Mathematica

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Let X be a normed space and $G_{F} (X)$ the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if $G_{F} (X)$ acts transitively on the unit sphere then X must be an inner product space.

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 $ℝ^{n}$ . 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 $ℓ_{p}$ -spaces in $ℝ^{n}$ for $0 < p < 1$ .

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 $(ℝ, | | \cdot | |_{)}$ be a Minkowski space with a unit ball and let $ϱ_{H}^{}$ be the Hausdorff metric induced by ${| | \cdot | |}_{}$ 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 $ϱ_{H}^{B ⁿ}$ for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace $(², ϱ_{H}^{})$ over any two-dimensional Minkowski space.

$ω$ -compact elements of normed algebras

Gh. Constantin (1973)

Matematički Vesnik

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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 $ℝ^{n}$ . It is proved that if $n \geq 2$ , 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 $1 \leq p \leq \infty$ . It is also demonstrated...

Counting triangles that share their vertices with the unit $n$ -cube

Brandts, Jan, Cihangir, Apo

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This paper is about $0 / 1$ -triangles, which are the simplest nontrivial examples of $0 / 1$ -polytopes: convex hulls of a subset of vertices of the unit $n$ -cube $I^{n}$ . We consider the subclasses of right $0 / 1$ -triangles, and acute $0 / 1$ -triangles, which only have acute angles. They can be explicitly counted and enumerated, also modulo the symmetries of $I^{n}$ .

The AR-Property of the spaces of closed convex sets

Katsuro Sakai, Masato Yaguchi (2006)

Colloquium Mathematicae

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Let $C o n v_{H} (X)$ , $C o n v_{A W} (X)$ and $C o n v_{W} (X)$ be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of $C o n v_{H} (X)$ and the space $C o n v_{A W} (X)$ are AR. In case X is separable, $C o n v_{W} (X)$ is locally path-connected.

Inequalities of the Kahane-Khinchin type and sections of $L_{p}$ -balls

A. Koldobsky, A. Pajor, V. Yaskin (2008)

Studia Mathematica

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We extend Kahane-Khinchin type inequalities to the case p > -2. As an application we verify the slicing problem for the unit balls of finite-dimensional spaces that embed in $L_{p}$ , p > -2.

Product property for capacities in $ℂ^{N}$

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: $C_{ν} (E ₁ \times E ₂) = m i n (C_{ν ₁} (E ₁), C_{ν ₂} (E ₂))$ , where $E_{j}$ and $ν_{j}$ are respectively a compact set and a norm in $ℂ^{N_{j}}$ (j = 1,2), and ν is a norm in $ℂ^{N ₁ + N ₂}$ , ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of $ℂ^{N}$ , 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...

Generalizing the Johnson-Lindenstrauss lemma to k-dimensional affine subspaces

Alon Dmitriyuk, Yehoram Gordon (2009)

Studia Mathematica

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Let ε > 0 and 1 ≤ k ≤ n and let ${W_{l}}_{l = 1}^{p}$ be affine subspaces of ℝⁿ, each of dimension at most k. Let $m = O (ε^{- 2} (k + l o g p))$ if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map $H : ℝ ⁿ \to ℝ^{m}$ such that for all 1 ≤ l ≤ p and $x, y \in W_{l}$ we have ||x-y||₂ ≤ ||H(x)-H(y)||₂ ≤ (1+ε)||x-y||₂, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε,k,p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces...

Volumetric invariants and operators on random families of Banach spaces

Piotr Mankiewicz, Nicole Tomczak-Jaegermann (2003)

Studia Mathematica

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The geometry of random projections of centrally symmetric convex bodies in $ℝ^{N}$ is studied. It is shown that if for such a body K the Euclidean ball $B ₂^{N}$ is the ellipsoid of minimal volume containing it and a random n-dimensional projection $B = P_{H} (K)$ is “far” from $P_{H} (B ₂^{N})$ then the (random) body B is as “rigid” as its “distance” to $P_{H} (B ₂^{N})$ permits. The result holds for the full range of dimensions 1 ≤ n ≤ λN, for arbitrary λ ∈ (0,1).

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 $n \times n$ positive definite matrices $A$ and $B$ . Some geodesical convexity results with respect to the Riemannian structure of the $n \times n$ 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 characterization of sets in $ℝ^{2}$ with DC distance function

Dušan Pokorný, Luděk Zajíček (2022)

Czechoslovak Mathematical Journal

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We give a complete characterization of closed sets $F \subset ℝ^{2}$ whose distance function $d_{F} : = dist (\cdot, F)$ is DC (i.e., is the difference of two convex functions on $ℝ^{2}$ ). Using this characterization, a number of properties of such sets is proved.

On the differential geometry of some classes of infinite dimensional manifolds

Maysam Maysami Sadr, Danial Bouzarjomehri Amnieh (2024)

Archivum Mathematicum

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Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space $Γ_{X}$ of any manifold $X$ . The name comes from the fact that various elements of the geometry of $Γ_{X}$ are constructed via lifting of the corresponding elements of the geometry of $X$ . In this note, we construct a general algebraic framework for lifted geometry which can be applied to various “infinite dimensional spaces” associated to $X$ . In order to define...

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 $C o n v_{F} (ℝ ⁿ)$ be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that $C o n v_{F} (ℝ ⁿ) \approx ℝ ⁿ \times Q$ for every n > 1 whereas $C o n v_{F} (ℝ) \approx ℝ \times$ .

Best constants for the isoperimetric inequality in quantitative form

Marco Cicalese, Gian Paolo Leonardi (2013)

Journal of the European Mathematical Society

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We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$ -dimensional case, our main contribution is a method for determining the optimal coefficients $c_{1}, ..., c_{m}$ in the inequality $δ P (E) \geq \sum_{k = 1}^{m} c_{k} α {(E)}^{k} + o (α {(E)}^{m})$ , valid for each Borel set $E$ with positive and finite area, with $δ P (E)$ and $α (E)$ being, respectively, the $𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡$ and the $𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦$ of $E$ . In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of $𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠$ including the lower semicontinuous extension of $\frac{δ P (E)}{α {(E)}^{2}}$ , we...

Superapproximation of the partial derivatives in the space of linear triangular and bilinear quadrilateral finite elements

Dalík, Josef

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A method for the second-order approximation of the values of partial derivatives of an arbitrary smooth function $u = u (x_{1}, x_{2})$ in the vertices of a conformal and nonobtuse regular triangulation $𝒯_{h}$ consisting of triangles and convex quadrilaterals is described and its accuracy is illustrated numerically. The method assumes that the interpolant $Π_{h} (u)$ in the finite element space of the linear triangular and bilinear quadrilateral finite elements from $𝒯_{h}$ is known only.

Convex integration with constraints and applications to phase transitions and partial differential equations

Stefan Müller, Vladimír Šverák (1999)

Journal of the European Mathematical Society

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We study solutions of first order partial differential relations $D u \in K$ , where $u : Ω \subset ℝ^{n} \to ℝ^{m}$ is a Lipschitz map and $K$ is a bounded set in $m \times n$ matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of $D u$ and second we replace Gromov’s $P$ −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our...

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), $I_{D} [S] = ⋃_{x, y \in S} I_{D} [x, y]$ . 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...

Simplices rarely contain their circumcenter in high dimensions

Jon Eivind Vatne (2017)

Applications of Mathematics

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Acute triangles are defined by having all angles less than $π / 2$ , and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension $n \geq 3$ , acuteness is defined by demanding that all dihedral angles between $(n - 1)$ -dimensional faces are smaller than $π / 2$ . However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property...

Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions

Albert Baernstein II, Robert C. Culverhouse (2002)

Studia Mathematica

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Let $X = \sum_{i = 1}^{k} a_{i} U_{i}$ , $Y = \sum_{i = 1}^{k} b_{i} U_{i}$ , where the $U_{i}$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_{i}, b_{i}$ are real constants. We prove that if $b ²_{i}$ is majorized by $a ²_{i}$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L ² - L^{p}$ Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...