Polytopes and the mean value property
type: Item
instance of: scholarly article
title: Polytopes and the mean value property
zbMATH Open document ID: 0872.39014
DOI: 10.1007/BF02770872
author: Juan-Miguel Gracia
published in: Discrete \& Computational Geometry
publication date: 1997-09-16T00:00:00Z
review text: As main result the author offers an answer to an open problem posed by \textit{A. Friedman} and \textit{W. Littman} [Trans. Am. Math. Soc. 102, 167-180 (1962; Zbl 0103.32201)]: Let \(P\) be any finite union of closed convex solid polytopes in the \(n\)-dimensional real space. Then the set of all continuous functions, assuming at each point their mean value with respect to the union of those faces of \(P\) whose dimension is at most \(k\), is a finite dimensional linear space of polynomials (of harmonic polynomials if \(P\) satisfies certain symmetry properties).
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 28A25
Mathematics Subject Classification ID: 35G15
zbMATH DE Number: 994027
zbMATH Keywords: mean value property
zbMATH Keywords: closed convex solid polytopes
zbMATH Keywords: linear space of polynomials
zbMATH Keywords: harmonic polynomials
MaRDI profile type: Publication
cites work: A Mean Value Property of the Derivative of Quadratic Polynomials-without Mean Values and Derivatives
cites work: General and regular solutions of functional equations characterizing harmonic polynomials
cites work: An invariant volume-mean-value property
cites work: The volume mean-value property of harmonic functions
cites work: A harmonic quadrature formula characterizing open strips
cites work: Q3228839
cites work: A Mean-Value Property of Cubic Polynomials-without Mean Values
cites work: Mean-Values and Harmonic Polynomials
cites work: Regular solids and harmonic polynomials
cites work: Functions satisfying the mean value property in the limit
cites work: The ''three-squares'' theorem for continuous functions
cites work: A new look at interpolation theory for entire functions of one variable
cites work: Interpolation problems in \(C^ n\) with applications to harmonic analysis
cites work: Q3964695
cites work: Functions Satisfying a Weighted Average Property
cites work: A Mean Value Property of Elliptic Equations with Constant Coefficients
cites work: A Converse of the Volume-Mean-Value Property for Invariantly Harmonic Functions
cites work: On Darboux and Mean Value Properties
cites work: Q5787302
cites work: A Note on Complex Polynomials Having Rolle's Property and the Mean Value Property for Derivatives
cites work: Shorter Notes: On the Mean-Value Property of Harmonic Functions
cites work: On the mean-value property of harmonic functions
cites work: Mean value properties for symmetrically differentiable functions
cites work: Functions Having the Restricted Mean Value Property
cites work: On the Restricted Mean Value Property
cites work: Q3276845
cites work: Functions with a Mean Value Property II
cites work: On Polynomials Characterized by a Certain Mean Value Property
cites work: Basic sets of invariants for finite reflection groups
cites work: Invariants of Finite Reflection Groups and Mean Value Problems II
cites work: Functions Satisfying the Mean Value Property for Product Measures
cites work: Functions satisfying a discrete mean value property
cites work: Invariants of Finite Reflection Groups and Mean Value Problems
cites work: Regular Polytopes and Harmonic Polynomials
cites work: Mean-values and polyharmonic polynomials
cites work: Functions Satisfying the Mean Value Property
cites work: Bodies for Which Harmonic Functions Satisfy the Mean Value Property
cites work: On the mean value property of finely harmonic and finely hyperharmonic functions
cites work: Q4002278
cites work: A Note On the Mean Value Property
cites work: On Functions Satisfying the Mean Value Property with Respect to a Product Measure
cites work: On the Mean Value Property of Harmonic Functions and Best Harmonic L 1 -Approximation
cites work: On the Inverse mean value Property of Harmonic Functions on Strips
cites work: On the Mean-Value Property of Harmonic Functions
cites work: Q3035569
cites work: On the Algebraic Independence of Symmetric Functions
cites work: Mean values and harmonic functions
cites work: Inverse mean value property of harmonic functions
cites work: Volume Densities with the Mean Value Property for Harmonic Functions
cites work: On an Integral Mean-Value Theorem in Analytic Function Theory
cites work: On a Relation between the “Square” Functional Equation And The “Square” Mean-Value Property
cites work: On two functional equations connected with a mean-value property of polynomials
cites work: Four different unknown functions satisfying the triangle mean value property for harmonic polynomials
cites work: On the theorem of S. Kakutani, M. Nagumo and J. L. Walsh for the mean value property of harmonic and complex polynomials
cites work: On the mean value property of harmonic and complex polynomials
cites work: Four different unknown functions satisfying the triangle mean value property for harmonic polynomials, II
cites work: Q3831259
cites work: On the quasiarithmetic mean in a mean value property and the associated functional equation
cites work: Q3947818

Polytopes and the mean value property (cites work)
cites work: Boundedness on a set of positive measure and the mean value property characterizes polynomials on a space \(V^ n\)

Polytopes and the mean value property (cites work)
cites work: The Converse of Pólya’s Mean Value Theorem

Polytopes and the mean value property (cites work)
cites work: A Converse to the Mean Value Property on Homogeneous Trees

Polytopes and the mean value property (cites work)
cites work: Harmonic polynomials

Polytopes and the mean value property (cites work)
cites work: Remarks on a paper by A. Friedman

Polytopes and the mean value property (cites work)
cites work: Q4277665

Polytopes and the mean value property (cites work)
cites work: Q4306489

Polytopes and the mean value property (cites work)
cites work: Q3681115

Polytopes and the mean value property (cites work)
cites work: Functions Satisfying a Mean Value Property at their Zeros

Polytopes and the mean value property (cites work)
cites work: Differential Equations Invariant Under Finite Reflection Groups

Polytopes and the mean value property (cites work)
cites work: A mean value theorem for polynomials and harmonic polynomials

Polytopes and the mean value property (cites work)
cites work: On a generalized asymptotic mean value property

Polytopes and the mean value property (cites work)
cites work: Fonctions définies dans le plan et vérifiant certaines propriétés de moyenne

Polytopes and the mean value property (cites work)
cites work: Mean values and differential equations

Polytopes and the mean value property (cites work)
cites work: Offbeat Integral Geometry