scholarly journals About every convex set in any generic Riemannian manifold

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander Lytchak ◽  
Anton Petrunin
Keyword(s):  
Riemannian Manifold ◽  
Convex Set ◽  
Convex Hypersurface ◽  
Necessary Condition ◽  
Strictly Convex ◽  
Smooth Hypersurface

Abstract We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary, we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold ( M , g ) {(M,g)} . For example, if a convex set in ( M , g ) {(M,g)} is bounded by a smooth hypersurface, then it is strictly convex.

Prescribing Centro-Affine Curvature From One Convex Body to Another

2020 ◽  
Author(s):  
Alina Stancu
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Curvature Flow ◽  
Constant Volume ◽  
Convex Hypersurface ◽  
Strictly Convex ◽  
Initial Hypersurface ◽  
Affine Curvature ◽  
Convex Hypersurfaces

Abstract We study a curvature flow on smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^n$, which commutes with the action of $SL(n)$. The flow shrinks the initial hypersurface to a point that, if rescaled to enclose a domain of constant volume, is a smooth, closed, strictly convex hypersurface in $\mathbb{R}^n$ with centro-affine curvature proportional, but not always equal, to the centro-affine curvature of a fixed hypersurface. We outline some consequences of this result for the geometry of convex bodies and the logarithmic Minkowski inequality.

scholarly journals Rigidity of complex convex divisible sets

2018 ◽  
Vol 10 (04) ◽  
pp. 817-851
Author(s):  
Andrew M. Zimmer
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Discrete Group ◽  
Convex Sets ◽  
Convex Set ◽  
Real Projective Space ◽  
Open Convex ◽  
Strictly Convex ◽  
Complex Projective

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts cocompactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have [Formula: see text] boundary, and have word hyperbolic dividing group. In this paper we study a notion of convexity in complex projective space and show that the only divisible complex convex sets with [Formula: see text] boundary are the projective balls.

A Note on Hyperconvexity in Riemannian Manifolds

1959 ◽  
Vol 11 ◽  
pp. 576-582
Author(s):  
Albert Nijenhuis
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Shortest Path ◽  
Riemannian Manifolds ◽  
Convex Set ◽  
Geodesic Segment ◽  
Positive Definite ◽  
Classical Theorem ◽  
Arc Length ◽  
Class C

Let M denote a connected Riemannian manifold of class C3, with positive definite C2 metric. The curvature tensor then exists, and is continuous.By a classical theorem of J. H. C. Whitehead (1), every point x of M has the property that all sufficiently small spherical neighbourhoods V of x are convex; that is, (i) to every y,z ∈ V there is one and only one geodesic segment yz in M which is the shortest path joining them:f:([0, 1]) → M,f(0) = y, f(1) = z; and (ii) this segment yz lies entirely in V:f([0, 1]) V; (iii) if f is parametrized proportional to arc length, then f(t) is a C2 function of y, t, and z.Let V be a convex set in M; and let y1 y2, Z1, z2 ∈ V.

On Certain Classes of Irregular Languages

2020 ◽  
pp. 30-43
Author(s):  
A.I. Belousov ◽  
R.S. Ismagilov ◽  
L.E. Filippova
Keyword(s):  
Regular Language ◽  
Affine Space ◽  
Formal Languages ◽  
Finite Union ◽  
Finite Alphabet ◽  
Necessary Condition ◽  
Strictly Convex ◽  
The Difference ◽  
Distribution Vector ◽  
Distribution Vectors

Objective of this paper is to prove certain regularity and irregularity conditions in languages determined by a set of integer vectors called distribution vectors of the number of letters in words over a finite alphabet. Each language over the finite alphabet uniquely determines its proprietary set of distribution vectors and vice versa, i.e., each set of vectors is associated with a language having this set of distribution vectors. A single necessary condition for the language regularity was considered associated with the concept of Z+-plane (sets of points with non-negative integer coordinates lying on a plane in the affine space). The condition is that a set of distribution vectors determined by any regular language could be represented as a finite union of the Z+-planes. Certain sufficient irregularity conditions associated with the distribution vector properties were proven. Based on this, classes of irregular languages could be identified. These classes are determined by a set of vectors (points) that could not be represented as a finite union of the Z+-planes; by a set of vectors containing vectors with arbitrarily high values of each coordinate and having certain restrictions on the difference between maximum and minimum values of the coordinates; by a set of vectors called the sparse sets. A method is proposed for building such sets using strictly convex and strictly increasing numerical sequences. These sufficient irregularity conditions are based on the Myhill --- Nerode theorem, which is known in the formal languages' theory. Examples of applying the proved theorems to the analysis of languages' regularity/irregularity are presented

A necessary geometric condition for the existence of certain homoclinic orbits

1986 ◽  
Vol 100 (3) ◽  
pp. 591-594 ◽  
Author(s):  
J. F. Toland
Keyword(s):  
Convex Set ◽  
Homoclinic Orbits ◽  
Geometric Condition ◽  
Strictly Convex

Throughout this brief note V denotes a fixed, smooth, real-valued function on such that there exists a bounded, open, strictly convex set C with (the boundary of C), V > 0 in C, V = 0 on and V′(0) = 0.

The Infancy of Solar-Type Stars and its Relevance for the Origin of Life

1997 ◽  
Vol 161 ◽  
pp. 267-282 ◽  
Author(s):  
Thierry Montmerle
Keyword(s):  
Main Sequence ◽  
Solar Nebula ◽  
Circumstellar Disks ◽  
T Tauri Stars ◽  
Necessary Condition ◽  
Sequence Evolution ◽  
T Tauri ◽  
Solar Type ◽  
Optical Domain ◽  
The Origin Of Life

AbstractFor life to develop, planets are a necessary condition. Likewise, for planets to form, stars must be surrounded by circumstellar disks, at least some time during their pre-main sequence evolution. Much progress has been made recently in the study of young solar-like stars. In the optical domain, these stars are known as «T Tauri stars». A significant number show IR excess, and other phenomena indirectly suggesting the presence of circumstellar disks. The current wisdom is that there is an evolutionary sequence from protostars to T Tauri stars. This sequence is characterized by the initial presence of disks, with lifetimes ~ 1-10 Myr after the intial collapse of a dense envelope having given birth to a star. While they are present, about 30% of the disks have masses larger than the minimum solar nebula. Their disappearance may correspond to the growth of dust grains, followed by planetesimal and planet formation, but this is not yet demonstrated.

Universal ESEM

1993 ◽  
Vol 51 ◽  
pp. 786-787
Author(s):  
G.D. Danilatos
Keyword(s):  
Electron Microscope ◽  
Scanning Electron Microscope ◽  
High Vacuum ◽  
Natural Extension ◽  
Necessary Condition ◽  
Environmental Scanning ◽  
Detection Systems ◽  
Small Gain ◽  
Detection Medium ◽  
Scanning Electron

The environmental scanning electron microscope (ESEM) has evolved as the natural extension of the scanning electron microscope (SEM), both historically and technologically. ESEM allows the introduction of a gaseous environment in the specimen chamber, whereas SEM operates in vacuum. One of the detection systems in ESEM, namely, the gaseous detection device (GDD) is based on the presence of gas as a detection medium. This might be interpreted as a necessary condition for the ESEM to remain operational and, hence, one might have to change instruments for operation at low or high vacuum. Initially, we may maintain the presence of a conventional secondary electron (E-T) detector in a "stand-by" position to switch on when the vacuum becomes satisfactory for its operation. However, the "rough" or "low vacuum" range of pressure may still be considered as inaccessible by both the GDD and the E-T detector, because the former has presumably very small gain and the latter still breaks down.

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