Open Convex Cone Research Articles

Sharp weighted log-Sobolev inequalities: Characterization of equality cases and applications

By using optimal mass transport theory, we provide a direct proof to the sharp L p L^p -log-Sobolev inequality ( p ≥ 1 ) (p\geq 1) involving a log-concave homogeneous weight on an open convex cone E ⊆ R n E\subseteq \mathbb R^n . The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the L p L^p -log-Sobolev inequality. The characterization of the equality cases is new for p ≥ n p\geq n even in the unweighted setting and E = R n E=\mathbb R^n . As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases.

Stochastic modelling of symmetric positive definite material tensors

Spatial symmetries and invariances play an important role in the behaviour of materials and should be respected in the description and modelling of material properties. The focus here is the class of physically symmetric and positive definite tensors, as they appear often in the description of materials, and one wants to be able to prescribe certain classes of spatial symmetries and invariances for each member of the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly ‘higher’ spatial invariance class. We formulate a modelling framework which not only respects these two requirements—positive definiteness and invariance—but also allows a fine control over orientation on one hand, and strength / size on the other. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear space of physically symmetric tensors, we consider it advantageous to widen the notion of mean to the so-called Fréchet mean on a metric space, which is based on distance measures or metrics between positive definite tensors other than the usual Euclidean one. It is shown how the random ensemble can be modelled and generated, independently in its scaling and orientational or directional aspects, with a Lie algebra representation via a memoryless transformation. The parameters which describe the elements in this Lie algebra are then to be considered as random fields on the domain of interest. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties—scaling, orientation, and prescribed material symmetry—on the desired quantities of interest.

Non-homogeneous Riemannian gradient equations for sum of squares of Bures–Wasserstein metric

We investigate Riemannian gradient equations on the open convex cone Pm of all m×m positive definite Hermitian matrices, for the function WA(X)=∑j=1nwjdW2(X,Aj)where dW denotes the Bures–Wasserstein metric. On the special case where the gradient of the weighted sum of squares of the Bures–Wasserstein metrics vanishes, its unique solution is known as the Wasserstein mean of A1,…,An. We discuss the existence and uniqueness of solutions for non-homogeneous Riemannian gradient equations at which vector fields are constant and congruence transformations. Furthermore, we establish some bounds for these solutions.

A note on the real support of Silva–Hasumi–Morimoto ultrahyperfunctions

Recently, we have studied the relation between the singular spectrum of a class of tempered ultrahyperfunctions corresponding to proper convex cones and their expressions as boundary values of holomorphic functions. In this note, we added some more results on the singular spectrum of the tempered ultrahyperfunctions. More specifically, the “microlocalization” of a Silva–Hasumi–Morimoto ultrahyperfunction corresponding to an open proper convex cone is addressed. Following Nishiwada’s approach to the analytic wave front set, we introduce the real singular spectrum of such an ultrahyperfunction and prove that it canonically projects onto Silva’s real support.

Mixture and interpolation of the parameterized ordered means

Loewner partial order plays a very important role in metric topology and operator inequality on the open convex cone of positive invertible operators. In this paper, we consider a family of the ordered means for positive invertible operators equipped with homogeneity and properties related to the Loewner partial order such as the monotonicity, joint concavity, and arithmetic-G-harmonic weighted mean inequalities. Similar to the resolvent average, we construct a parameterized ordered mean and compare two types of mixtures of parameterized ordered means in terms of the Loewner order. We also show a relation between two families of parameterized ordered means associated with the power mean monotonic interpolating given two parameterized ordered means.

Smooth dependence of fixed points of Hamiltonians

We consider the Lie semigroup of symplectic Hamiltonians acting on the open convex cone of positive definite matrices via linear fractional transformations. Each member of its interior contracts strictly the invariant Finsler metric, the Thompson metric on the cone, and has a unique positive definite fixed point. We show that the fixed point map is smooth. As applications, we obtain the smooth dependence of the solutions of discrete algebraic Riccati equations and a family of smooth maps from the Siegel upper half-plane over the cone of positive definite matrices into its imaginary part.

Generalized Vector-Valued Hardy Functions

We consider analytic functions in tubes Rn+iB⊂Cn with values in Banach space or Hilbert space. The base of the tube B will be a proper open connected subset of Rn, an open connected cone in Rn, an open convex cone in Rn, and a regular cone in Rn, with this latter cone being an open convex cone which does not contain any entire straight lines. The analytic functions satisfy several different growth conditions in Lp norm, and all of the resulting spaces of analytic functions generalize the vector valued Hardy space Hp in Cn. The analytic functions are represented as the Fourier–Laplace transform of certain vector valued Lp functions which are characterized in the analysis. We give a characterization of the spaces of analytic functions in which the spaces are in fact subsets of the Hardy functions Hp. We obtain boundary value results on the distinguished boundary Rn+i{0¯} and on the topological boundary Rn+i∂B of the tube for the analytic functions in the Lp and vector valued tempered distribution topologies. Suggestions for associated future research are given.

Sobolev inequalities with jointly concave weights on convex cones

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form ∫ E | u ( x ) | q ω ( x ) d x 1 / q ⩽ K 0 ∫ E | ∇ u ( x ) | p σ ( x ) d x 1 / p , u ∈ C 0 ∞ ( R n ) , (WSI)where p ⩾ 1 and q > 0 is the corresponding Sobolev critical exponent. Here E ⊆ R n is an open convex cone, and ω , σ : E → ( 0 , ∞ ) are two homogeneous weights verifying a general concavity-type structural condition. The constant K 0 = K 0 ( n , p , q , ω , σ ) > 0 is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that K 0 is optimal in (WSI) if and only if ω and σ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to partial differential equations are also provided.

The general linear equation on open connected sets

Fix non-zero reals \(\alpha _1\), \(\ldots , \)\(\alpha _n\) with \(n\ge 2\) and let \(K\) be a non-empty open connected set in a topological vector space such that \(\sum _{i\le n}\alpha _iK\subseteq K\) (which holds, in particular, if \(K\) is an open convex cone and \(\alpha _1,\ldots ,\alpha _n>0\)). Let also \(Y\) be a vector space over \(\mathbb{F}\it :=\mathbb{Q} \it (\alpha _1,\ldots ,\alpha _n)\). We show, among others, that a function \(f : K\rightarrow Y\) satisfies the general linear equation $$\forall x_1,\ldots, x_n \in K, \quad f\big (\sum _{i\le n}\alpha _i x_i\big )=\sum _{i\le n}\alpha _i f(x_i)$$ if and only if there exist a unique \(\mathbb{F}\it \)-linear \(A X \rightarrow Y\) and unique \(b\in Y\) such that \(f(x)=A(x)+b\) for all \(x \in K\), with \(b=0\) if \(\sum _{i\le n}\alpha _i\ne 1\). The main tool of the proof is a general version of a result Rado and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.

A Note on Discrete Einstein Metrics

In this note, we prove that the space of all admissible piecewise linear metrics parameterized by the square of length on a triangulated manifold is a convex cone. We further study Regge’s Einstein-Hilbert action and give a more reasonable definition of discrete Einstein metric than the former version in [3]. Finally, we introduce a discrete Ricci flow for three dimensional triangulated manifolds, which is closely related to the existence of discrete Einstein metrics. 1 The space of piecewise linear metrics Consider an n dimensional compact manifold M with a triangulation T . The triangulation is written as T = {T0,T1, · · · ,Tn}, where Ti (0 ≤ i ≤ n) represents the set of all i dimensional simplices. A piecewise linear metric is a map l : T1 → (0,+∞) making each simplex an Euclidean simplex. There are two disadvantages to think of l as the analogue of smooth Riemannian metric tensor g. For one thing, we know that MT , the space of all admissible piecewise linear metrics, is not convex (although it is a simply connected open set). For another, the scaling property of l is not good enough. If the smooth Riemannian metric tensor g scales to cg in the smooth manifold M, then the length l(γ) of a curve γ : [0, 1] → M scales to √ cl(γ). If we take l2 as the direct analogue of metric tensor g, both the above two disadvantages can be overcome. The idea of considering the square of l, not l itself, as an analogue of smooth Riemannian metric tensor comes naturally from the former work by the first author and Xu [4], where the idea has been used for piecewise linear manifolds with circle or sphere packing metrics. Firstly, we have Theorem 1.1. For manifold M with triangulation T , denote gij = l2 ij for each adjacent edge i ∼ j. Then MT , the space of all admissible piecewise linear metrics parameterized by gij , is a nonempty connected open convex cone.

The Order Properties and Karcher Barycenters of Probability Measures on the Open Convex Cone

We study the probability measures on the open convex cone of positive definite operators equipped with the Loewner ordering. We show that two crucial push-forward measures derived by the congruence transformation and power map preserve the stochastic order for probability measures. By the continuity of two push-forward measures with respect to the Wasserstein distance, we verify several interesting properties of the Karcher barycenter for probability measures with finite first moment such as the invariant properties and the inequality for unitarily invariant norms. Moreover, the characterization for the stochastic order of uniformly distributed probability measures has been shown.

On Rank One Convex Functions that are Homogeneous of Degree One

We show that positively $1$--homogeneous rank one convex functions are convex at $0$ and at matrices of rank one. The result is a special case of an abstract convexity result that we establish for positively $1$--homogeneous directionally convex functions defined on an open convex cone in a finite dimensional vector space. From these results we derive a number of consequences including various generalizations of the Ornstein $\LL^1$ non inequalities. Most of the results were announced in ({\em C.~R.~Acad.~Sci.~Paris, Ser.~I 349 (2011), 407--409}).

On additive correspondences

Abstract The existence of additive selections of additive correspondences was investigated in [Ark. Mat. 4 (1960), 87–97], [Rev. Roumaine Math. Pures Appl. 28 (1983), 239–242.], [Math. Ser. Univ. Novi Sad 18 (1988), 143–148] and other papers. In this article, we find an existence theorem for additive selections of additive correspondences with convex compact values in a real normed linear space defined on an open convex cone of a real separable normed space.

POLARIZED REAL TORI

For a fixed positive integer g, we let <TEX>$\mathcal{P}_g=\{Y{\in}\mathbb{R}^{(g,g)}{\mid}Y=^tY</TEX><TEX>&</TEX><TEX>gt;0\}$</TEX> be the open convex cone in the Euclidean space <TEX>$\mathbb{R}^{g(g+1)/2}$</TEX>. Then the general linear group GL(g, <TEX>$\mathbb{R}$</TEX>) acts naturally on <TEX>$\mathcal{P}_g$</TEX> by <TEX>$A{\star}Y=AY^tA(A{\in}GL(g,\mathbb{R}),\;Y{\in}\mathcal{P}_g)$</TEX>. We introduce a notion of polarized real tori. We show that the open cone <TEX>$\mathcal{P}_g$</TEX> parametrizes principally polarized real tori of dimension g and that the Minkowski modular space 𝔗g = <TEX>$GL(g,\mathbb{Z}){\backslash}\mathcal{P}_g$</TEX> may be regarded as a moduli space of principally polarized real tori of dimension g. We also study smooth line bundles on a polarized real torus by relating them to holomorphic line bundles on its associated polarized real abelian variety.

Discrete automorphism groups of convex cones of finite type

AbstractWe investigate subgroups of $\text{SL}(n,\mathbb{Z})$ which preserve an open nondegenerate convex cone in $\mathbb{R}^{n}$ and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on self-dual cones, Weyl groups of certain Kac–Moody algebras, and they do occur in algebraic geometry as the automorphism groups of projective manifolds acting on their ample cones.

Nash-type equilibria on Riemannian manifolds: A variational approach

Motivated by Nash equilibrium problems on ‘curved’ strategy sets, the concept of Nash–Stampacchia equilibrium points is introduced via variational inequalities on Riemannian manifolds. Characterizations, existence, and stability of Nash–Stampacchia equilibria are studied when the strategy sets are compact/noncompact geodesic convex subsets of Hadamard manifolds, exploiting two well-known geometrical features of these spaces both involving the metric projection map. These properties actually characterize the non-positivity of the sectional curvature of complete and simply connected Riemannian spaces, delimiting the Hadamard manifolds as the optimal geometrical framework of Nash–Stampacchia equilibrium problems. Our analytical approach exploits various elements from set-valued and variational analysis, dynamical systems, and non-smooth calculus on Riemannian manifolds. Examples are presented on the Poincaré upper-plane model and on the open convex cone of symmetric positive definite matrices endowed with the trace-type Killing form.

A CHARACTERIZATION OF ELLIPTIC HYPERBOLOIDS

Consider a non-degenerate open convex cone C with vertex the origin in the <TEX>$n$</TEX>2-dimensional Euclidean space <TEX>$E^n$</TEX>. We study volume properties of strictly convex hypersurfaces in the cone C. As a result, for example, if the volume of the region of an elliptic cone C cut off by the tangent hyperplane P of M at <TEX>$p$</TEX> is independent of the point <TEX>$p{\in}M$</TEX>, then it is shown that the hypersurface M is part of an elliptic hyperboloid.

Asymptotic Dependence for Light-Tailed Homothetic Densities

Dependence between coordinate extremes is a key factor in any multivariate risk assessment. Hence, it is of interest to know whether the components of a given multivariate random vector exhibit asymptotic independence or asymptotic dependence. In the latter case the structure of the asymptotic dependence has to be clarified. In the multivariate setting it is common to have an explicit form of the density rather than the distribution function. In this paper we therefore give criteria for asymptotic dependence in terms of the density. We consider distributions with light tails and restrict attention to continuous unimodal densities defined on the whole space or on an open convex cone. For simplicity, the density is assumed to behomothetic: all level sets have the same shape. Balkema and Nolde (2010) contains conditions on the shape which guarantee asymptotic independence. The situation for asymptotic dependence, treated in the present paper, is more delicate.

Asymptotic Dependence for Light-Tailed Homothetic Densities

Dependence between coordinate extremes is a key factor in any multivariate risk assessment. Hence, it is of interest to know whether the components of a given multivariate random vector exhibit asymptotic independence or asymptotic dependence. In the latter case the structure of the asymptotic dependence has to be clarified. In the multivariate setting it is common to have an explicit form of the density rather than the distribution function. In this paper we therefore give criteria for asymptotic dependence in terms of the density. We consider distributions with light tails and restrict attention to continuous unimodal densities defined on the whole space or on an open convex cone. For simplicity, the density is assumed to be homothetic: all level sets have the same shape. Balkema and Nolde (2010) contains conditions on the shape which guarantee asymptotic independence. The situation for asymptotic dependence, treated in the present paper, is more delicate.

Causal structure and electrodynamics on Finsler spacetimes

We present a concise new definition of Finsler spacetimes that generalize Lorentzian metric manifolds and provide consistent backgrounds for physics. Extending standard mathematical constructions known from Finsler spaces we show that geometric objects like the Cartan non-linear connection and its curvature are well-defined almost everywhere on Finsler spacetimes, also on their null structure. This allows us to describe the complete causal structure in terms of timelike and null curves; these are essential to model physical observers and the propagation of light. We prove that the timelike directions form an open convex cone with null boundary as is the case in Lorentzian geometry. Moreover, we develop action integrals for physical field theories on Finsler spacetimes, and tools to deduce the corresponding equations of motion. These are applied to construct a theory of electrodynamics that confirms the claimed propagation of light along Finsler null geodesics.