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Convex polytopes whose projection bodies and difference sets are polars

Horst Martini
1991 Discrete & Computational Geometry  
The aim of this paper is to show that a convex d-polytope (d -> 3) is a simplex if and only if its projection body and its difference set are polars.  ...  In a paper by the author and B. Weissbach it was proved that the projection body and the difference set of a d-simplex (d -> 2) are polars.  ...  (B) The difference set DP and the projection body liP are polars with respect to a sphere 6. S d-~ for some 6 ~ R +. (C) The inner 1-quermass and the brightness of P satisfy y,(P, u).  ... 
doi:10.1007/bf02574676 fatcat:lgmttesnb5hqdd43cyuarcgrqq

Page 5639 of Mathematical Reviews Vol. , Issue 86m [page]

1986 Mathematical Reviews  
I. 86m:52003 Stability in Aleksandrov’s problem for a convex body, one of whose projections is a ball. (Russian) Ukrain. Geom. Sb. No. 28 (1985), 50-62, ii.  ...  Suppose A is a convex body in Euclidean n-space whose support function is twice continuously differentiable away from the origin and, further, that the boundary of A has positive principal radii of curvature  ... 

The Classification of Regular Solids

T. M. Madden, S. A. Robertson
1995 Bulletin of the London Mathematical Society  
In [4] , Farran and Robertson extended the classical concept of regularity from convex polytopes to convex bodies in general.  ...  A convex body that is regular in this new sense is called a regular solid. Thus the set ^ of all regular polytopes is a subset of the set 5^ of all regular solids.  ...  A convex body that is regular in this new sense is called a regular solid. Thus the set ^ of all regular polytopes is a subset of the set 5^ of all regular solids.  ... 
doi:10.1112/blms/27.4.363 fatcat:zwlqtwtrznbhto2junyos23f3q

Gorenstein Fano toric degenerations [article]

Christian Steinert
2020 arXiv   pre-print
For the proof of this statement we will study polytopes whose polar dual is a lattice polytope.  ...  Additionally, we conjecture a necessary and sufficient condition for the Ehrhart quasi-polynomial of a rational convex polytope to be a polynomial.  ...  A rational convex polytope is called reflexive if the polytope itself and its polar dual are lattice polytopes (i.e. they have integral vertices).  ... 
arXiv:2011.12591v1 fatcat:vysj56bbufgp5adqvxwffeoyhe

Page 5599 of Mathematical Reviews Vol. , Issue 97I [page]

1997 Mathematical Reviews  
In Sections 5—6, the mean projections and the affine surface-area of Busemann of the difference body are investigated.  ...  In this theory convex sets are replaced by star-shaped sets and support functions by radial functions.  ... 

The nonlinear geometry of linear programming. III.\ Projective Legendre transform coordinates and Hilbert geometry

J. C. Lagarias
1990 Transactions of the American Mathematical Society  
This paper studies projective scaling trajectories, which are the trajectories obtained by following the infinitesimal version of Karmarkar's linear programming algorithm.  ...  The projective Legendre transform mapping has a coordinate-free geometric interpretation in  ...  With this labelling the dual polytope P~ (defined in §2B) is identified with the polar polytope P~, where the polar body CO to a convex body C containing o in its interior is defined by CO = {y: (x, y)  ... 
doi:10.1090/s0002-9947-1990-1058199-0 fatcat:3teei74k2jee3niaot4dsqyhle

On the volume of the John-Löwner ellipsoid [article]

Grigory Ivanov
2017 arXiv   pre-print
Also, we describe all possible vectors in ^n, whose coordinates are the squared lengths of a projection of the standard basis in ^n onto a k-dimensional subspace.  ...  We find an optimal upper bound on the volume of the John ellipsoid of a k-dimensional section of the n-dimensional cube, and an optimal lower bound on the volume of the Löwner ellipsoid of a projection  ...  I am grateful for Marton Naszodi for useful remarks and help with the text. I am especially grateful for Roma Karasev for fruitful conversations concerned the topic of this paper.  ... 
arXiv:1707.04442v2 fatcat:aynaehbk5rburdd6u6bkkd2dlq

The Nonlinear Geometry of Linear Programming. III Projective Legendre Transform Coordinates and Hilbert Geometry

J. C. Lagarias
1990 Transactions of the American Mathematical Society  
This paper studies projective scaling trajectories, which are the trajectories obtained by following the infinitesimal version of Karmarkar's linear programming algorithm.  ...  The projective Legendre transform mapping has a coordinate-free geometric interpretation in  ...  With this labelling the dual polytope P~ (defined in §2B) is identified with the polar polytope P~, where the polar body CO to a convex body C containing o in its interior is defined by CO = {y: (x, y)  ... 
doi:10.2307/2001758 fatcat:6v6edxrd6rdofi7x5mf53eqyam

Page 7569 of Mathematical Reviews Vol. , Issue 96m [page]

1996 Mathematical Reviews  
Billera and Munson have proved that not every matroid polytope has a polar, i.e. the convex hull of the vertices of a polytope and the intersection of the supporting half-spaces have different generalizations  ...  Summary: “Sets of points are called separable if their convex hulls are disjoint. We suggest a technique for optimal partitioning of a set N into two separable subsets, Nj, N2.  ... 

Polytope Approximation and the Mahler Volume [chapter]

Sunil Arya, Guilherme D. da Fonseca, David M. Mount
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
The problem of approximating convex bodies by polytopes is an important and well studied problem.  ...  This is a dimensionless quantity that involves the product of the volumes of a convex body and its polar dual.  ...  Clearly, polar r (K) is a scaled copy of polar(K) by a factor of r 2 . The bodies polar √ ε (K) and Γ q are bounded by the same set of halfspaces, and so we have: Lemma 3.3.  ... 
doi:10.1137/1.9781611973099.3 dblp:conf/soda/AryaFM12 fatcat:rbxoll52r5c43copiuog3jz43e

Page 2721 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews  
(+x1,---,+Xn)/2, where Q, is the convex closure of the set {+x\,---,+X,} from bdC and d2, measures the density of that set in a special way. There is a polar version of this second result.  ...  A polarity argument gives the same estimate when Y, is replaced by the class of polytopes that have at most 2n facets and are symmetric with respect to the origin.  ... 

Page 5529 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews  
K denotes a convex body in d-dimensional Euclidean space whose volume centroid is y. K|A denotes the orthogonal projection of K onto a (d —k)-dimensional subspace A.  ...  5529 93j:52011 52A39 52A40 Spingarn, Jonathan E. (1-GAIT) An inequality for sections and projections of a convex set. (English summary) Proc. Amer. Math. Soc. 118 (1993), no. 4, 1219-1224.  ... 

Spectrahedral Lifts of Convex Sets [article]

Rekha R. Thomas
2018 arXiv   pre-print
The main result is that projection representations of a convex set are controlled by factorizations, through closed convex cones, of an operator that comes from the convex set.  ...  Efficient representations of convex sets are of crucial importance for many algorithms that work with them.  ...  I am indebted to all my collaborators on the projects that contributed to this paper. I thank Pablo Parrilo for the construction in Example 1.5 and for several useful conversations.  ... 
arXiv:1803.08079v1 fatcat:t4zjo5nbpzdw3eldn4d6ldlo2i

The computational complexity of convex bodies [article]

Alexander Barvinok, Ellen Veomett
2006 arXiv   pre-print
We discuss approximations of a convex body by an ellipsoid, by an algebraic hypersurface, by a projection of a polytope with a controlled number of facets, and by a section of the cone of positive semidefinite  ...  We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: "given an x in V, does x belong to X?"  ...  Nemirovski for explaining a way to tightly approximate the Euclidean ball by the projection of a polytope with not too many facets (Section 4.4) and for encouragement.  ... 
arXiv:math/0610325v1 fatcat:o7bvjht2nnbvpg3mtg4vve6nnu

Page 5295 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
One source of examples are certain “Wythoffian” polytopes (and their duals), whose vertex-sets are orbits of points under a finite reflection group or its rotation sub- group.  ...  The only examples are volume, volume of the polar body and the Euler characteristic. The existence of the so-called L,-affine surface areas [see E. Lutwak, Adv. Math.  ... 
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