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Nonlinear elliptic equations, rearrangements of functions and orlicz spaces

This paper gives a review of Frechet-bounds and their applications. In section two an approach to the marginal problem and Frechet-bounds based on duality theory resp. the Hahn-Banach theorem is discussed. Main applications concern the Strassen representation theorem for stochastic orders, the sharpness of the classical Frechet-bounds, the representation of minimal metrics, couplings of distributions, the Monge-Kantorovic-problem, the construction of random variables with maximum (resp. minimum) sum and variances of the sum, maximally dependent random variables and others. For multivariate marginal systems there is a useful reduction principle and there are some bounds for simple systems, which yield a characterization of the marginal problem for a system of two dimensional marginals in a three-fold product space. In section three we discuss some generalizations of the Younginequality, which are useful for solving the dual problems of the Frechet-bounds. A basic notion in this connection is the notion of c-convex functions. As an application one can give a nice characterization of solutions of certain transportation problems. We give a probabilistic proof of some generalizations of the Young- and the Oppenheim-inequality. In section four we discuss some statistical applications and problems. The Huzurbazar conjecture on marginal sufficiency, the problem of the optimal combination of marginal tests and the question of estimation theory in marginal models is considered.

FréChet-Bounds and Their Applications

Publisher Summary
This chapter discusses special convex bodies like simplices, ellipsoids, centrally symmetric bodies, and bodies of constant width. A simplex in Ed is defined as the convex hull of d + 1 affinely independent points. Leichtweiss used the infimum of the quotients of circumradius and inradius over classes of affinely equivalent convex bodies, Santalo and Eggleston used the area and a certain width integral of a plane convex body in order to characterize simplices. For studying ellipsoids from the view point of convexity one has a good starting point in the surveys of Bonnesen and Fenchel, Gruber and Hobinger, and Petty, which summarize results on characterizations of ellipsoid by means of sections and projections; by extremal properties; using the group of projectivities or subgroups thereof; and also on ellipsoids in Minkowski geometry and Hilbert geometry. Furthermore, there are a lot of geometric inequalities yielding ellipsoids as extremal bodies. Also, approximation of convex bodies by convex polytopes may lead to ellipsoids as extremal bodies. Ellipsoids may also be characterized by the product of mean width and surface area or by the quotient of inradius and circumradius. Characterizations of Euclidean spaces or, in infinite dimensions, of inner product spaces also give characterizations of ellipsoids. Also, Klee and Bolker have characterized centrally symmetric polytopes by intersection and projection properties.

CHAPTER 1.11 – Special Convex Bodies

http://link.springer.com/content/pdf/10.1007/978-94-009-8555-1_12.pdf

Lorenz Ordering Within the Generalized Gamma Family of Income Distributions

The “traffic-assignment problem” consists of predicting “Wardrop-equilibrium flows” on a roadway network when origin-to-destination “input flows” are specified. The “demand-equilibrium problem” consists of predicting those input flows that place the network in a state of “economic equilibrium” when the input flows are related via given travel-demand (feedback) curves to the resulting Wardrop-equilibrium origin-to-destination “travel costs”.

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Traffic Equilibria Analysed via Geometric Programming.

(1970). An Analytic Proof of Young's Inequality. The American Mathematical Monthly: Vol. 77, No. 6, pp. 603-609.

J. B. Diaz

Papers

An Analytic Proof of Young's Inequality

A Note on Primes, with Arbitrary Initial or Terminal Decimal Ciphers, in Dirichlet Arithmetic Progressions