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The {\em minimal unfolded region} (or the {\em heart}) of a bounded subset $\Om$ in the Euclidean space is a subset of the convex hull of $\Om$ the definition of which is based on reflections in hyperplanes. It was introduced to restrict the location of the points that give extreme values of certain functions, such as potentials whose kernels are monotone functions of the distance, and solutions of differential equations to which Aleksandrov's reflection principle can be applied. %the temperature of a heat conductor with some initial-boundary condition, in which case the points are called the hot spots. We show that the minimal unfolded regions of the convex hull and parallel bodies of $\Om$ are both included in that of $\Om$., Comment: to appear in Hokkaido Math. J

In this note, we establish some new interesting inequalities in Ptolemaic spaces.

In this paper, using Bottema's inequality for two triangles and other results, the generalization of an inequality involving the medians and angle-bisectors of the triangle is proved. This settles affirmatively a problem posed by J-Liu.

In this paper, a class of geometric inequalities for the volumes of two $n-$simplexes and their $k-$subsimplexes are established. The results are generalizations to several dimensions of the well-known Neuberg-Pedoe inequality of two triangles.

A four part dissection and rearrangement provides a new proof of the isoperimetric inequality in the plane as well as a new approach to Bonnesen-type error estimates for the isoperimetric deficit of compact convex sets and of star bodies that are centrally symmetric with respect to the origin. An isoperimetric inequality in R bounds the area (or related functional) of a compact set by some function of the perimeter (or related functional) of that same set. The classical isoperimetric inequality asserts that, for any compact subset S ⊆ R having well-defined perimeter

We show how 'test' vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger's constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger's constant in terms of the inradius of a plane domain., 9 pages, added references 6,11,30, to appear in Archiv der Math

In the paper we consider an extension of Mobius-Pompeiu theorem of the elementary geometry over metric spaces. We specially take into consideration Ptolemaic metric spaces., Version without pictures. Minor corrections in examples

International audience; The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp “weighted isoperimetric inequality” and a nonsharp classical inequality similar to the one obtained by J. Michael and L. Simon. The proof relies on the description of a solution of the problem of Monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension.