Refinements of generalized triangle inequalities

In this paper, we discuss refinements of the well-known triangle inequality and it is reverse inequality for strongly integrable functions with values in a Banach space X. We also discuss refinement of a generalized triangle inequality of the second kind for Lp functions. For both cases, the attainability of the equality is also investigated.     

Convex rank 1 subsets of Euclidean buildings (of type A 2)

For a Euclidean building X of type A ₂, we classify the 0-dimensional subbuildings A of ∂ _T X that occur as the asymptotic boundary of closed convex subsets. In particular, we show that triviality of the holonomy of a triple (of points of A) is (essentially) sufficient. To prove this, we construct new convex subsets as the union of convex sets.      

Another approach to characterizations of generalized triangle inequalities in normed spaces

In this paper, we consider a generalized triangle inequality of the following type: $$\left\| {x_1 + \cdots + x_n } \right\|^p \leqslant \frac{{\left\| {x_1 } \right\|^p }} {{\mu _1 }} + \cdots + \frac{{\left\| {x_2 } \right\|^p }} {{\mu _n }}\left( {for all x_1 , \ldots ,x_n \in X} \right),$$ where (X, ‖·‖) is a normed space, (µ₁, ..., µ _n ) ∈ ? ⁿ and p > 0. By using ψ-direct sums of Banach spaces, we present another approach to characterizations of the above inequality which is given by [Dadipour F., Moslehian M.S., Rassias J.M., Takahasi S.-E., Nonlinear Anal., 2012, 75(2), 735–741].     

Some inequalities in inner product spaces related to the generalized triangle inequality

In this paper we obtain some inequalities related to the generalized triangle and quadratic triangle inequalities for vectors in inner product spaces. Some results that employ the Ostrowski discrete inequality for vectors in normed linear spaces are also obtained.     

Coarse equivalences of Euclidean buildings

We prove the following rigidity results. Coarse equivalences between metrically complete Euclidean buildings preserve spherical buildings at infinity. If all irreducible factors have dimension at least two, then coarsely equivalent Euclidean buildings are isometric (up to scaling factors); if in addition none of the irreducible factors is a Euclidean cone, then the isometry is unique and has finite distance from the coarse equivalence.     

Stability inequalities and universal Schubert calculus of rank 2

The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each ??sufficiently rich?? spherical building Y of type W we associate a certain cohomology theory $ H_{BK}^*(Y) $ and verify that, first, it depends only on W (i.e., all such buildings are ??homotopy equivalent??), and second, $ H_{BK}^*(Y) $ is the associated graded of the coinvariant algebra of W under certain filtration. We also construct the dual homology ??pre-ring?? on Y. The convex ??stability?? cones in $ {\left( {{\mathbb{R}^2}} \right)^m} $ defined via these (co)homology theories of Y are then shown to solve the problem of classifying weighted semistable m-tuples on Y in the sense of [KLM1]; equivalently, they are cut out by the generalized triangle inequalities for thick Euclidean buildings with the Tits boundary Y. The independence of the (co)homology theory of Y refines the result of [KLM2], which asserted that the Stability Cone depends on W rather than on Y. Quite remarkably, the cohomology ring $ H_{BK}^*(Y) $ is obtained from a certain universal algebra A _t by a kind of ??crystal limit?? that has been previously introduced by Belkale?CKumar for the cohomology of ag varieties and Grassmannians. Another degeneration of A _t leads to the homology theory H _*(Y).     

On sharp triangle inequalities in Banach spaces

We shall present a couple of norm inequalities which will much improve the sharp triangle inequality with n elements and its reverse inequality in a Banach space shown recently by the last three authors.     

Isoperimetric inequalities for lattices in semisimple lie groups of rank 2

LetG be a connected semisimple Lie group with finite center. Let be an irreducible non-uniform lattice inG. We show that if the real rank ofG is 2, then the Dehn (or filling) function of is exponential.     

The center conjecture for thick spherical buildings

In this paper we show that a convex subcomplex of a spherical building of type E ₆, E ₇ or E ₈ is a subbuilding or the automorphisms of the subcomplex fix a a point on it. Together with previous results of Mühlherr–Tits, and Leeb and the author, this completes the proof of Tits’ Center Conjecture for spherical buildings without factors of type H ₄, in particular, for thick spherical buildings.     

Isoperimetric inequalities of Euclidean type in metric spaces

No Abstract. .Received: August 2003 Revision: June 2004 Accepted: August 2004     

Generalization of the triangle and Ptolemy inequalities

The Euclidean triangle inequality generalizes to an alternating inequality for any oddsided polygon that can be inscribed in a circle; there is equality in the even cases. A generalization of Ptolemy's theorem follows by inversion. The results have Minkowskian analogues.     

On the nonnegative rank of Euclidean distance matrices

The Euclidean distance matrix for n distinct points in R^r is generically of rank r + 2. It is shown in this paper via a geometric argument that its nonnegative rank for the case r = 1 is generically n.     

Some norm inequalities concerning generalized inverses, 2

We extend to the von Neumann–Schatten classes C_p and norms ·_p, where 2  p < ∞, Penrose’s result on minimizing AXB − C₂. We give an example to show that this extension does not hold for 1  p < 2. The proof of the global inequality depends on local considerations.     

Rellich type inequalities in domains of the Euclidean space

For smooth functions supported in a domain of the Euclidean space we investigate two Rellich type inequalities with weights which are powers of the distance function. We prove that for an arbitrary plane domain there exist positive Rellich constants in these inequalities if and only if the boundary of the domain is a uniformly perfect set. Moreover, we obtain explicit estimates of constants in function of geometric domain characteristics. Also, we find sharp constants in these Rellich type inequalities for all non-convex domains of dimension d ≥ 2 provided that the domains satisfy the exterior sphere condition with certain restriction on the radius of spheres.     

Linear preservers of extremes of rank inequalities over semirings: Term rank and zero-term rank

We characterize linear operators on matrices over semirings that preserve the extremal cases in the bounds on term and zero-term ranks of sums and products of matrices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 3–21, 2004.     

Flat rank of automorphism groups of buildings

The flat rank of a totally disconnected locally compact group G, denoted flat-rk(G), is an invariant of the topological group structure of G. It is defined thanks to a natural distance on the space of compact open subgroups of G. For a topological Kac-Moody group G with Weyl group W, we derive the inequalities alg-rk(W) ≤ flat-rk(G) ≤ rk(|W|₀). Here, alg-rk(W) is the maximal Z-rank of abelian subgroups of W, and rk(|W|₀) is the maximal dimension of isometrically embedded flats in the CAT0-realization |W|₀. We can prove these inequalities under weaker assumptions. We also show that for any integer n ≥ 1 there is a simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G) = n and which is not linear.     

广义Baskakov型算子的强逆不等式

利用二阶Ditzian-Totik模考虑一类算子的强逆不等式，这类不等式曾经被许多者用不同的方法研究过。本文将采用一种统一的方法来处理一大类算子的强逆不等式，得到了它们的Ditzian-Ivanov结果。本方法适用于更广的算子。