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

FUNCTIONS OF MARKOV PROCESSES AND ALGEBRAIC MEASURES

1992 ◽  
Vol 04 (01) ◽  
pp. 39-64 ◽  
Author(s):  
M. FANNES ◽  
B. NACHTERGAELE ◽  
L. SLEGERS
Keyword(s):  
Markov Processes ◽  
Invariant Measures ◽  
Canonical Representation ◽  
Positive Matrix ◽  
Translation Invariant ◽  
State Spaces ◽  
Finite State ◽  
The Mean ◽  
Class Of Functions ◽  
Translation Invariant Measures

We introduce a class of translation-invariant measures on the set {0, …, q−1}ℤ determined by a set of q d-dimensional matrices. They are algebraic in the sense that their densities are obtained by applying a functional to products of the defining matrices. Positivity of probabilities is assured by assuming a positivity structure on the algebra of defining matrices. Restricting attention to the usual positivity notion of positive matrix elements, a detailed analysis leads to a canonical representation theorem that solves the parametrization problem. Furthermore, we show that the class of algebraic measures coincides with the class of functions of Markov processes with finite state spaces. Our main result consists in the detailed study of the asymptotics of the conditional probabilities from which we derive a formula for the mean entropy.

Translation invariant measures which are not Hausdorff measures

1966 ◽  
Vol 62 (4) ◽  
pp. 693-698 ◽  
Author(s):  
K. E. Hirst
Keyword(s):  
Invariant Measures ◽  
Hausdorff Measures ◽  
Translation Invariant ◽  
Translation Invariant Measures

An important and much-investigated class of measures is the class of Hausdorff measures, first defined by Hausdorff (1). These measures form a subclass of the class of translation invariant measures, but just how wide a class they form is not known.

scholarly journals Existence and uniqueness of translation invariant measures in separable Banach spaces

2014 ◽  
Vol 50 (2) ◽  
pp. 401-419 ◽  
Author(s):  
Tepper Gill ◽  
Aleks Kirtadze ◽  
Gogi Pantsulaia ◽  
Anatolij Plichko
Keyword(s):  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Invariant Measures ◽  
Translation Invariant ◽  
Translation Invariant Measures

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.

Monotone translation invariant valuations on convex bodies

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