Monotone translation invariant valuations on convex bodies

1990 ◽  
Vol 55 (6) ◽  
pp. 595-598 ◽  
Author(s):  
Peter McMullen
Keyword(s):  
Convex Bodies ◽  
Translation Invariant

scholarly journals Valuations on convex functions and convex sets and Monge–Ampère operators

2019 ◽  
Vol 19 (3) ◽  
pp. 313-322 ◽  
Author(s):  
Semyon Alesker
Keyword(s):  
Convex Functions ◽  
Convex Sets ◽  
Convex Bodies ◽  
Linear Functionals ◽  
Translation Invariant ◽  
Infinite Dimensional ◽  
Dense Image ◽  
Monge Ampere ◽  
Explicit Relation ◽  
Class Of Functions

Abstract The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

Determination of Boolean models by densities of mixed volumes

2019 ◽  
Vol 51 (01) ◽  
pp. 116-135
Author(s):  
Daniel Hug ◽  
Wolfgang Weil
Keyword(s):  
Uniqueness Theorem ◽  
Representation Theorem ◽  
Convex Bodies ◽  
Boolean Models ◽  
Mixed Volumes ◽  
Shape Information ◽  
Translation Invariant ◽  
Mean Values ◽  
Particle Process

AbstractIn Weil (2001) formulae were proved for stationary Boolean models Z in ℝd with convex or polyconvex grains, which express the densities (specific mean values) of mixed volumes of Z in terms of related mean values of the underlying Poisson particle process X. These formulae were then used to show that in dimensions 2 and 3 the densities of mixed volumes of Z determine the intensity γ of X. For d = 4, a corresponding result was also stated, but the proof given was incomplete, since in the formula for the density of the Euler characteristic V̅0(Z) of Z a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result has recently been proved by Hug, Rataj, and Weil (2013), (2018) and is based on flag measures of the convex bodies involved. Here, we show that such flag representations not only lead to a correct derivation of the four-dimensional result, but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter we make use of Alesker’s representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the nonstationary case.

Translation-invariant measures of planes intersecting convex bodies

1988 ◽  
Vol 43 (6) ◽  
pp. 2838-2847
Author(s):  
G. Yu. Panina
Keyword(s):  
Invariant Measures ◽  
Convex Bodies ◽  
Translation Invariant ◽  
Translation Invariant Measures

Continuous translation invariant valuations on convex bodies

1984 ◽  
Vol 54 (1) ◽  
pp. 95-105 ◽  
Author(s):  
U Betke ◽  
P. R Goodey
Keyword(s):  
Convex Bodies ◽  
Translation Invariant

scholarly journals Correction to: Continuous translation invariant valuations on convex bodies

1986 ◽  
Vol 56 (1) ◽  
pp. 253-253
Author(s):  
U. Betke ◽  
P. R. Godey
Keyword(s):  
Convex Bodies ◽  
Translation Invariant

Convex Bodies: The Brunn–Minkowski Theory

1993 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Convex Bodies

Translation-Invariant Gibbs Measures for the Blum-Kapel Model on a Cayley Tree

2016 ◽  
Vol 15 (2) ◽  
pp. 239-255 ◽  
Author(s):  
Nosir Khatamov ◽  
◽  
Rustam Khakimov ◽  
Keyword(s):  
Cayley Tree ◽  
Gibbs Measures ◽  
Translation Invariant

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

Excess entropy of the systems of molecules differing in size and shape

1983 ◽  
Vol 48 (1) ◽  
pp. 192-198 ◽  
Author(s):  
Tomáš Boublík
Keyword(s):  
Equation Of State ◽  
Hard Spheres ◽  
Excess Entropy ◽  
Convex Bodies ◽  
Pure Substance ◽  
Liquid Equilibrium ◽  
Simple Rules ◽  
Different Shapes ◽  
The Mean

The excess entropy of mixing of mixtures of hard spheres and spherocylinders is determined from an equation of state of hard convex bodies. The obtained dependence of excess entropy on composition was used to find the accuracy of determining ΔSE from relations employed for the correlation and prediction of vapour-liquid equilibrium. Simple rules were proposed for establishing the mean parameter of nonsphericity for mixtures of hard bodies of different shapes allowing to describe the P-V-T behaviour of solutions in terms of the equation of state fo pure substance. The determination of ΔSE by means of these rules is discussed.

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