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### Fundamentals of Restricted-Orientation Convexity

Eugene Fink, Derick Wood
1996 Information Sciences
O-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] .  ...

### Fundamentals of Restricted-Orientation Convexity

Eugene Fink, Derick Wood
2018
A 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] .  ...

### A theorem on convex surfaces

Chin-shui Hs{ü
1962 Proceedings of the American Mathematical Society
Suppose S and S are closed 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  ...

### Closed convex hypersurfaces with second fundamental form of constant curvature

Rolf Schneider
1972 Proceedings of the American Mathematical Society
An 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  ...

### Closed Convex Hypersurfaces with Second Fundamental Form of Constant Curvature

Rolf Schneider
1972 Proceedings of the American Mathematical Society
An 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  ...

### The Equilateral Pentagon at Zero Angular Momentum: Maximal Rotation through Optimal Deformation

William Tong, Holger R. Dullin
2012 SIAM Journal on Applied Dynamical Systems
We also show that 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.  ...

### Restricted-orientation half-spaces

Eugene Fink, Derick Wood, Robert A. Melter, Angela Y. Wu, Longin J. Latecki
1996 Vision Geometry V
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.  ...  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.  ...

### Locally convex hypersurfaces of negatively curved spaces

S. Alexander
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.  ...

### Locally Convex Hypersurfaces of Negatively Curved Spaces

S. Alexander
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.  ...

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

Feng Luo
1999 arXiv   pre-print
This paper attempts to relate some ideas 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  ...

### Affine Conormal of Convex Hypersurfaces

Chi-Ming Yau
1989 Proceedings of the American Mathematical Society
The geometry 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.  ...

### Affine conormal of convex hypersurfaces

Chi-Ming Yau
1989 Proceedings of the American Mathematical Society
The geometry 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.  ...

### Some Uses of the Second Conformal Structure on Strictly Convex Surfaces

Tilla Klotz
1963 Proceedings of the American Mathematical Society
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.  ...  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.  ...