Compact Riemannian Manifolds with Homogeneous Geodesics

A homogeneous Riemannian space $(M= G/H,g)$ is called a geodesic orbit space (shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of the isometry group $G$. We study the structure of compact GO-spaces and give some sufficient conditions for existence and non-existence of an invariant metric $g$ with homogeneous geodesics on a homogeneous space of a compact Lie group $G$. We give a classification of compact simply connected GO-spaces $(M = G/H,g)$ of positive Euler characteristic. If the group $G$ is simple and the metric $g$ does not come from a bi-invariant metric of $G$, then $M$ is one of the flag manifolds $M_1=SO(2n+1)/U(n)$ or $M_2= Sp(n)/U(1)\cdot Sp(n-1)$ and $g$ is any invariant metric on $M$ which depends on two real parameters. In both cases, there exists unique (up to a scaling) symmetric metric $g_0$ such that $(M,g_0)$ is the symmetric space $M = SO(2n+2)/U(n+1)$ or, respectively, $\mathbb{C}P^{2n-1}$. The manifolds $M_1$, $M_2$ are weakly symmetric spaces

One property of a planar curve whose convex hull covers a given convex figure

In this note, we prove the following conjecture by A. Akopyan and V. Vysotsky: If the convex hull of a planar curve $\gamma$ covers a planar convex figure $K$, then $\operatorname{length}(\gamma) \geq \operatorname{per} (K) - \operatorname{diam} (K)$. In addition, all cases of equality in this inequality are studied.Comment: 10 pages, 4 figure

Perfect and almost perfect homogeneous polytopes

The paper is devoted to perfect and almost perfect homogeneous polytopes in Euclidean spaces. We classified perfect and almost perfect polytopes among all regular polytopes and all semiregular polytopes excepting Archimedean solids and two four-dimensional Gosset polytopes. Also we construct some non-regular homogeneous polytopes that are (or are not) perfect and posed some unsolved questions.Comment: 18 pages, 2 figure

Legal Details of the International Organization INMARSAT

International Mobile Satellite Organization INMARSAT, established in 1976, has proven its viability and effectiveness by ensuring continuous interaction of states and national organizations in achieving common objectives of the creation and operation of its space segment for the purpose of providing commercial radio communication services for mobile sea, ground and air units, as well as non-commercial radio communication services, including services within the Global Maritime Distress Safety System (GMDSS). In this connection, it is of both academic and practical interest to study the details of INMARSAT’s international law nature. This international organisation is unique because, while having international legal personality, it is directly engaged in providing its space segment. This has affected the structure and content of INMARSAT’ foundation documents (Convention and Operational Agreement), the nature of its membership and content of its bodies’ functions, the rights and obligations of a member state and the national communications organisation nominated by it as the INMARSAT participant. An analysis of INMARSAT’s international law nature will allow to assess the extent to which its organizational and legal structure meets the requirements