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Convex polytopes whose projection bodies and difference sets are polars

1991
*
Discrete & Computational Geometry
*

The aim of this paper is to show that a

doi:10.1007/bf02574676
fatcat:lgmttesnb5hqdd43cyuarcgrqq
*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). ...##
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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 ...##
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The Classification of Regular Solids

1995
*
Bulletin of the London Mathematical Society
*

In [4] , Farran

doi:10.1112/blms/27.4.363
fatcat:zwlqtwtrznbhto2junyos23f3q
*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. ...##
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Gorenstein Fano toric degenerations
[article]

2020
*
arXiv
*
pre-print

For the proof of this statement we will study

arXiv:2011.12591v1
fatcat:vysj56bbufgp5adqvxwffeoyhe
*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). ...##
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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. ...##
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The nonlinear geometry of linear programming. III.\ Projective Legendre transform coordinates and Hilbert geometry

1990
*
Transactions of the American Mathematical Society
*

This paper studies

doi:10.1090/s0002-9947-1990-1058199-0
fatcat:3teei74k2jee3niaot4dsqyhle
*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) ...##
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On the volume of the John-Löwner ellipsoid
[article]

2017
*
arXiv
*
pre-print

Also, we describe all possible vectors in ^n,

arXiv:1707.04442v2
fatcat:aynaehbk5rburdd6u6bkkd2dlq
*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. ...##
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The Nonlinear Geometry of Linear Programming. III Projective Legendre Transform Coordinates and Hilbert Geometry

1990
*
Transactions of the American Mathematical Society
*

This paper studies

doi:10.2307/2001758
fatcat:6v6edxrd6rdofi7x5mf53eqyam
*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) ...##
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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. ...##
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Polytope Approximation and the Mahler Volume
[chapter]

2012
*
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
*

The problem of approximating

doi:10.1137/1.9781611973099.3
dblp:conf/soda/AryaFM12
fatcat:rbxoll52r5c43copiuog3jz43e
*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. ...##
###
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. ...##
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Spectrahedral Lifts of Convex Sets
[article]

2018
*
arXiv
*
pre-print

The main result is that

arXiv:1803.08079v1
fatcat:t4zjo5nbpzdw3eldn4d6ldlo2i
*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. ...##
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The computational complexity of convex bodies
[article]

2006
*
arXiv
*
pre-print

We discuss approximations of a

arXiv:math/0610325v1
fatcat:o7bvjht2nnbvpg3mtg4vve6nnu
*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. ...##
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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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