Related papers: The Pompeiu problem

Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}$, where $\mathcal{S}$ is the Schwartz class of distributions, and $$\int_{\sigma (D)} f(x) dx = 0 \quad \forall \sigma \in G, \qquad (*)$$ where $D\subset \R^n$ is a bounded domain, the closure…

Assume that $D\subset \mathbb{R}^3$ is a bounded domain with $C^1-$smooth boundary. Our result is: {\bf Theorem 1.} {\em If $D$ has $P-$property, then $D$ is a ball.} Four equivalent formulations of the Pompeiu problem are discussed. A…

The Pompeiu problem is considered as shape optimization problem. We show stability of the ball which is the minimum point of related domain functional. The proof is based on shape derivative method. Stability of the ball for general domain…

Let $K$ be a compact subgroup of a locally compact group $G$. We investigate a Pompeiu type problem for homogeneous spaces $G/K$. Suppose $E$ is a compact subset of $G/K$. Using recent work of L\'{a}szl\'{o} Sz\'{e}kelyhidi on $K$-spectral…

We consider a two-sided Pompeiu type problem for a discrete group $G$. We give necessary and sufficient conditions for a finite set $K$ of $G$ to have the $\mathcal{F}(G)$-Pompeiu property. Using group von Neumann algebra techniques, we…

We introduce a kind of converse of Pompeiu's theorem. Fix an equilateral triangle $\triangle A_0B_0C_0$, then for any triangle $\triangle ABC$ there is a unique point $P$ inside the circumcircle $\Gamma_0$ of $\triangle A_0B_0C_0$ such that…

We formulate a version of the Pompeiu problem in the discrete group setting. Necessary and sufficient conditions are given for a finite collection of finite subsets of a discrete abelian group, whose torsion free rank is less than the…

We say that a finite subset $E$ of the Euclidean plane $\mathbb{R}^2$ has the discrete Pompeiu property with respect to isometries (similarities), if, whenever $f:\mathbb{R}^2\to \mathbb{C}$ is such that the sum of the values of $f$ on any…

Let $B_{x}\subseteq\mathbb{R}^{n}$ denote the Euclidean ball with diameter $[0,x]$, i.e. with with center at $\frac{x}{2}$ and radius $\frac{\left|x\right|}{2}$. We call such a ball a petal. A flower $F$ is any union of petals, i.e.…

In this note we consider the following conjecture: given any closed symplectic manifold $M$, there is a sufficiently small real positive number $\rho$ such that the open ball of radius $\rho$ in the Hofer metric centered at the identity on…

A bounded domain $\Omega$ in a Riemannian manifold $M$ is said to have the Pompeiu property if the only continuous function which integrates to zero on $\Omega$ and on all its congruent images is the zero function. In some respects, the…

We investigate the Pompeiu property for subsets of the real line, under no assumption of connectedness. In particular we focus our study on finite unions of bounded (disjoint) intervals, and we emphasize the different results corresponding…

Given a fixed-point free compact holomorphic self-map $f$ on a bounded symmetric domain $D$, which may be infinite dimensional, we establish the existence of a family $\{H(\xi, \lambda)\}_{\lambda >0}$ of convex $f$-invariant domains at a…

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N+1}$ with a connected $C^{2,\epsilon}$ ($\epsilon\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha u\,\,…

We investigate the discrete Fuglede's conjecture and Pompeiu problem on finite abelian groups and develop a strong connection between the two problems. We give a geometric condition under which a multiset of a finite abelian group has the…

We consider a class of topological objects in the 3-sphere $S^3$ which will be called {\it $n$-punctured ball tangles}. Using the Kauffman bracket at $A=e^{\pi i/4}$, an invariant for a special type of $n$-punctured ball tangles is defined.…

The Deligne-Simpson problem (DSP) (resp. the weak DSP) is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ or $c_j\subset gl(n,{\bf C})$ so that there…

In this paper we present some results for a connected infinite graph $G$ with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of $G$. (For a vertex $w$ of a…

On a compact Riemann surface $\Sigma$ of genus $g>1$, equipped with a complex vector bundle $E$ of rank $2$ and degree zero let $M_H$ be the moduli space of Higgs bundles. $M_H$ admits a $\mathbb C^{\star}$-action and to each stable…

In this paper we discuss a couple of observations related to polynomial convexity. More precisely, (i) We observe that the union of finitely many disjoint closed balls with centres in $\cup_{\theta\in[0,\pi/2]}e^{i\theta}V$ is polynomially…