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Fundamentals of Restricted-Orientation Convexity

1996
*
Information Sciences
*

O-

doi:10.1016/0020-0255(96)00056-4
fatcat:hrvct4humncddgqzy5vahgbhbm
*convexity*, also called*restricted*-*orientation**convexity*, was defined in terms*of*the intersection*of*a geometric object with lines parallel to the elements*of*a fixed*orientation*set O (see Section 2 ... Rawlins applied*restricted**orientations*in his definition*of*two new types*of*generalized*convexity*, which he called O-*convexity*and strong O-*convexity*[16] . ... In two dimensions, if the*orientation*set contains n lines, the boundary*of*every O-*convex*polygon can be partitioned into at most n O-*convex*polygonal lines [16] . ...##
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Fundamentals of Restricted-Orientation Convexity

2018

A

doi:10.1184/r1/6605903.v1
fatcat:dakasvu6trgblgvq4zaozx6a3q
*restricted*-*orientation**convex*set, also called an O-*convex*set, is a set*of*points whose intersection with lines from some fixed set is empty or connected. ... We introduce and investigate*restricted*-*orientation*analogs*of*lines, flats, and hyperplanes, and characterize O-*convex*and O-connected sets in terms*of*their intersections with hyperplanes. ... In two dimensions, if the*orientation*set contains n lines, the boundary*of*every O-*convex*polygon can be partitioned into at most n O-*convex*polygonal lines [16] . ...##
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A theorem on convex surfaces

1962
*
Proceedings of the American Mathematical Society
*

Suppose S and S are closed

doi:10.1090/s0002-9939-1962-0137079-7
fatcat:qt2wfn4xhfd4jftcujkorquk5i
*orientable*strictly*convex*C2 surfaces and h : S->S is an order-preserving differentiable homeomorphism. ... , where hij and h{j are the second*fundamental*tensors, (hij) = (hi,)~l and g, g are the determinants*of*the first*fundamental*tensors, then h is a homothetic transformation. Proof. ... License or copyright*restrictions*may apply to redistribution; see http://www.ams.org/journal-terms-*of*-use ...##
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Closed convex hypersurfaces with second fundamental form of constant curvature

1972
*
Proceedings of the American Mathematical Society
*

An

doi:10.1090/s0002-9939-1972-0307133-1
fatcat:iocdgddj55apxastfemdqr5txu
*oriented*hypersurface (always assumed to be sufficiently smooth) in (/i+l)-dimensional Euclidean space (n^2) will be called closed if it is compact and without boundary, and*convex*if its second*fundamental*... A closed*convex*hypersurface on which the second*fundamental*form is*of*constant Riemannian curvature has to be a Euclidean sphere. ... License or copyright*restrictions*may apply to redistribution; see http://www.ams.org/journal-terms-*of*-use closed*convex*hypersurfaces License or copyright*restrictions*may apply to redistribution ...##
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Closed Convex Hypersurfaces with Second Fundamental Form of Constant Curvature

1972
*
Proceedings of the American Mathematical Society
*

An

doi:10.2307/2038475
fatcat:37xih7xzbvexlbhqeurtknz7a4
*oriented*hypersurface (always assumed to be sufficiently smooth) in (/i+l)-dimensional Euclidean space (n^2) will be called closed if it is compact and without boundary, and*convex*if its second*fundamental*... A closed*convex*hypersurface on which the second*fundamental*form is*of*constant Riemannian curvature has to be a Euclidean sphere. ... License or copyright*restrictions*may apply to redistribution; see http://www.ams.org/journal-terms-*of*-use closed*convex*hypersurfaces License or copyright*restrictions*may apply to redistribution ...##
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The Equilateral Pentagon at Zero Angular Momentum: Maximal Rotation through Optimal Deformation

2012
*
SIAM Journal on Applied Dynamical Systems
*

We also show that

doi:10.1137/110853868
fatcat:i7woc7qhbjdord5jvfqrpc3apq
*restricting*allowed shapes to*convex*pentagons the optimal loop is the boundary*of*the*convex*region and gives \Delta \theta \approx 19\degree. ... We show that the shape space*of*the equilateral pentagon has genus 4 and find a*fundamental*region by discrete symmetry reduction with respect to symmetry group D_5. ... HRD would like to thank the Department*of*Applied Mathematics*of*the University*of*Colorado at Boulder for their hospitality. ...##
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Restricted-orientation half-spaces

1996
*
Vision Geometry V
*

We i n troduce

doi:10.1117/12.251784
fatcat:n3leekrtzbds5myii3tj4zejlq
*restricted*-*orientation*halfspaces, which are an O-*convexity*analog*of*halfspaces, explore their properties, and demonstrate their relationship to*restricted*-*orientation**convex*sets. ... We i n troduce*restricted*-*orientation*halfspaces, which are an O-*convexity*analog*of*halfspaces, explore their properties, and demonstrate their relationship to*restricted*-*orientation**convex*sets. ... r a n t Committee*of*Hong Kong. ...##
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Locally convex hypersurfaces of negatively curved spaces

1977
*
Proceedings of the American Mathematical Society
*

*of*a

*convex*body. ... A well-known theorem due to Hadamard states that if the second

*fundamental*form

*of*a compact immersed hypersurface M

*of*Euclidean space E" (n > 3) is positive definite, then M is imbedded as the boundary ...

*of*a

*convex*body. ...

##
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Locally Convex Hypersurfaces of Negatively Curved Spaces

1977
*
Proceedings of the American Mathematical Society
*

*of*a

*convex*body. ... A well-known theorem due to Hadamard states that if the second

*fundamental*form

*of*a compact immersed hypersurface M

*of*Euclidean space E" (n > 3) is positive definite, then M is imbedded as the boundary ...

*of*a

*convex*body. ...

##
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Grothendieck's Reconstruction Principle and 2-dimensional Topology and Geometry
[article]

1999
*
arXiv
*
pre-print

This paper attempts to relate some ideas

arXiv:math/9904019v1
fatcat:5lxudvdasbcefg5d2zehosrhqm
*of*Grothendieck in his Esquisse d'un programme and some*of*the recent results on 2-dimensional topology and geometry. ... Especially, we shall discuss the Teichmüller theory, the mapping class groups, SL(2, C) representation variety*of*surface groups, and Thurston's theory*of*measured laminations. ... Namely, suppose f is a complex valued function defined on the*fundamental*group*of*the surface so that the*restriction**of*f to the*fundamental*group*of*each essential level-1 subsurface is a GL(n, C)-character ...##
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Affine Conormal of Convex Hypersurfaces

1989
*
Proceedings of the American Mathematical Society
*

The geometry

doi:10.2307/2048828
fatcat:yfgmxvi4b5g7degjdeeyb26go4
*of**convex*hypersurfaces in real affine space is analyzed using the affine conormal. A weak version*of*Chern's conjecture, characterizing paraboloids among*convex*graphs, is proved. ... In addition, it is shown that a closed*convex*affine hypersurface with constant affine total curvature is an ellipsoid. ... It removes the superfluous assumption on positive definiteness*of*the affine third*fundamental*form when the Theorem was first formulated. Thanks are also due to Profs. ...##
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Affine conormal of convex hypersurfaces

1989
*
Proceedings of the American Mathematical Society
*

The geometry

doi:10.1090/s0002-9939-1989-0965947-1
fatcat:r56kejxbp5d3jb3unxnx3dokbq
*of**convex*hypersurfaces in real affine space is analyzed using the affine conormal. A weak version*of*Chern's conjecture, characterizing paraboloids among*convex*graphs, is proved. ... In addition, it is shown that a closed*convex*affine hypersurface with constant affine total curvature is an ellipsoid. ... It removes the superfluous assumption on positive definiteness*of*the affine third*fundamental*form when the Theorem was first formulated. Thanks are also due to Profs. ...##
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Some Uses of the Second Conformal Structure on Strictly Convex Surfaces

1963
*
Proceedings of the American Mathematical Society
*

But if S is strictly

doi:10.2307/2034997
fatcat:tgy5ttvv5rcyzhwvqbwwi5j7da
*convex*(and*oriented*so that mean curvature 77 >0) then the second*fundamental*form is positive definite and determines still another conformai structure on S. ... We imitate Hopfs procedures in this paper,*restricting*our attention to strictly*convex*surfaces and using R2 in place*of*Pi structure. ... But if S is strictly*convex*(and*oriented*so that mean curvature 77 >0) then the second*fundamental*form is positive definite and determines still another conformai structure on S. ...##
###
Page 527 of Illinois Journal of Mathematics Vol. 4, Issue 4
[page]

1960
*
Illinois Journal of Mathematics
*

Suppose that there exists an

*orientation*-preserving diffeomorphism f*of*the manifold M,, onto the manifold M* such that, at each pair*of*corresponding points, the mani- folds M,, and M% have a common*fundamental*... This normal frame Xén41 +++ €n4m is called a*fundamental*normal frame*of*the star manifold M, at the point X. ...##
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Immersions of Positively Curved Manifolds Into Manifolds with Curvature Bounded Above

1990
*
Transactions of the American Mathematical Society
*

Then the radius

doi:10.2307/2001332
fatcat:q7euzroacffsfluvmcwmaukhhq
*of*the smallest A'-ball containing x(M) is less than ^nc and x is in fact an imbedding*of*M into A', whose image bounds a*convex*body. ... Let M be a compact, connected,*orientable*Riemannian manifold*of*dimension n-1 > 2 , and let x be an isometric immersion*of*M into an ndimensional Riemannian manifold N . ... The conditions on M in Theorem 1 are less*restrictive*than those*of*Theorem 2, where M is both*orientable*and*of*codimension 1. In Theorem 1, M need not be*orientable*and can be*of*any codimension. ...
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