This paper recalls the work of D. Pompeiu who introduced the notion of set distance in his thesis published one century ago. The notion was further studied by F. Hausdorff, C. Kuratowski who acknowledged in their books the contribution of Pompeiu and it is frequently called the Hausdorff distance. keywords: Hausdorff distance, Hausdorff-Pompeiu distance, Pompeiu functions, Pompeiu conjecture, S...

We demonstrate connections that exists between a conjecture of Schiffer’s (which is equivalent to a positive answer to the Pompeiu problem), stationary solutions to the Euler equations, and the convergence of solutions to the Navier-Stokes equations to that of the Euler equations in the limit as viscosity vanishes. We say that a domain Ω ⊆ R, d ≥ 2, has the Pompeiu property if, given that the i...

Several symmetry problems are discussed. These include the Pompeiu problem and similar conjectures for the heat and wave equations.

The aim of the present paper is to prove that the family of all closed nonempty subsets of a complete probabilistic metric space L is complete with respect to the probabilistic Pompeiu-Hausdorff metric H . The same is true for the families of all closed bounded, respectively compact, nonempty subsets of L. If L is a complete random normed space in the sense of Šerstnev, then the family of all n...

By studying the distances of a point to the sides, respectively the vertices of an equilateral triangle, certain new identities and inequalities are deduced. Some inequalities for the elements of the Pompeiu triangle are also established.

Let f ∈ Lloc(R) ∩ S ′, S ′ is the Schwartz class of distributions, and ∫ σ(D) f(x)dx = 0 ∀σ ∈ G, (∗) whereD ⊂ Rn, n ≥ 2, is a bounded domain, the closure D̄ of which is C1−diffeomorphic to a closed ball. Then the complement of D̄ is connected and path connected. Here G denotes the group of all rigid motions in Rn. This group consists of all translations and rotations. It is conjectured that if f ...

We consider a generalized Pompeiu equation in the space of Schwartz distributions and as an application we find the locally integrable solutions of the equation. © 2005 Elsevier Ltd. All rights reserved. MSC: 39B22; 47H15; 46F05

Thus for "most" choices of three squares /]1, P2, P3 the validity of (1) for -{P~, P2, P3} implies f ---0. The Pompeiu problem has an extensive history which we will not discuss here. The reader is referred to the paper [13] of L. ZALCMAN for an excellent account of the problem, including the "two-circles theorem" of Delsarte for harmonic functions, which is the motivation for our Theorem 1. We...