scholarly journals Index of Grassmann manifolds and orthogonal shadows

2018 ◽  
Vol 30 (6) ◽  
pp. 1539-1572
Author(s):  
Djordje Baralić ◽  
Pavle V. M. Blagojević ◽  
Roman Karasev ◽  
Aleksandar Vučić
Keyword(s):  
Vector Bundle ◽  
Convex Body ◽  
Wreath Product ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Orthogonal Complement ◽  
Grassmann Manifolds ◽  
The Real ◽  
Geometric Result

Abstract In this paper, we study the {\mathbb{Z}/2} action on the real Grassmann manifolds {G_{n}(\mathbb{R}^{2n})} and {\widetilde{G}_{n}(\mathbb{R}^{2n})} given by taking the (appropriately oriented) orthogonal complement. We completely evaluate the related {\mathbb{Z}/2} Fadell–Husseini index utilizing a novel computation of the Stiefel–Whitney classes of the wreath product of a vector bundle. These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For {n=2^{a}(2b+1)} , {k=2^{a+1}-1} , a convex body C in {\mathbb{R}^{2n}} , and k real-valued functions {\alpha_{1},\ldots,\alpha_{k}} continuous on convex bodies in {\mathbb{R}^{2n}} with respect to the Hausdorff metric, there exists a subspace {V\subseteq\mathbb{R}^{2n}} such that projections of C to V and its orthogonal complement {V^{\perp}} have the same value with respect to each function {\alpha_{i}} , that is, {\alpha_{i}(p_{V}(C))=\alpha_{i}(p_{V^{\perp}}(C))} for all {1\leq i\leq k} .

scholarly journals A note on the characteristic rank and related numbers

2014 ◽  
Vol 64 (6) ◽  
Author(s):  
Ľudovít Balko ◽  
Július Korbaš
Keyword(s):  
Vector Bundle ◽  
Topological Space ◽  
Betti Numbers ◽  
Sharp Inequality ◽  
Grassmann Manifolds ◽  
The Real ◽  
Partitions Of Integers

AbstractThis note quantifies, via a sharp inequality, an interplay between (a)the characteristic rank of a vector bundle over a topological space X (b)the ℤ2-Betti numbers of X, and(c)sums of the numbers of certain partitions of integers.In a particular context, (c) is transformed into a sum of the readily calculable Betti numbers of the real Grassmann manifolds.

scholarly journals Diameters in Typical Convex Bodies

1990 ◽  
Vol 42 (1) ◽  
pp. 50-61 ◽  
Author(s):  
Imre Bárány ◽  
Tudor Zamfirescu
Keyword(s):  
Convex Body ◽  
Convex Sets ◽  
Compact Convex ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
General Setting ◽  
Baire Space ◽  
Supporting Hyperplanes ◽  
The World

The most usual diameters in the world are those of a sphere and they all contain its centre. More generally, a chord of a convex body in Rd is called a diameter if there are two parallel supporting hyperplanes at the two endpoints of the chord.It is easily seen that there are points on at least two diameters. From a result of Kosiński [6] proved in a more general setting it follows that every convex body has a point lying on at least three diameters. Does a typical convex body behave like a sphere and contain a point on infinitely or even uncountably many diameters?But what is a typical convex body? The space 𝒦 of all convex bodies (d-dimensional compact convex sets) in Rd, equipped with the Hausdorff metric, is a Baire space.

scholarly journals Polytopal approximation of elongated convex bodies

2018 ◽  
Vol 18 (1) ◽  
pp. 105-114
Author(s):  
Gilles Bonnet
Keyword(s):  
Convex Body ◽  
Best Approximation ◽  
Convex Set ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
The Best Approximation ◽  
Polytopal Approximation ◽  
Isoperimetric Ratio

AbstractThis paper presents bounds for the best approximation, with respect to the Hausdorff metric, of a convex bodyKby a circumscribed polytopePwith a given number of facets. These bounds are of particular interest ifKis elongated. To measure the elongation of the convex set, its isoperimetric ratioVj(K)1/jVi(K)−1/iis used.

On Bodies Associated with a Given Convex Body

1996 ◽  
Vol 39 (4) ◽  
pp. 448-459 ◽  
Author(s):  
Endre Makai ◽  
Horst Martini
Keyword(s):  
Convex Body ◽  
Cross Section ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Shadow Boundary ◽  
Projection Body ◽  
Intersection Body ◽  
Open Set ◽  
Difference Body ◽  
The Difference

AbstractLet d ≥ 2, and K ⊂ ℝd be a convex body with 0 ∈ int K. We consider the intersection body IK, the cross-section body CK and the projection body ΠK of K, which satisfy IK ⊂ CK ⊂ ΠK. We prove that [bd(IK)] ∩ [bd(CK)] ≠ (a joint observation with R. J. Gardner), while for d ≥ 3 the relation [CK] ⊂ int(ΠK) holds for K in a dense open set of convex bodies, in the Hausdorff metric. If IK = c ˙ CK for some constant c > 0, then K is centred, and if both IK and CK are centred balls, then K is a centred ball. If the chordal symmetral and the difference body of K are constant multiples of each other, then K is centred; if both are centred balls, then K is a centred ball. For d ≥ 3 we determine the minimal number of facets, and estimate the minimal number of vertices, of a convex d-polytope P having no plane shadow boundary with respect to parallel illumination (this property is related to the inclusion [CP] ⊂ int(ΠP)).

scholarly journals Polytopes and $C^1$-convex bodies

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Karim Adiprasito ◽  
José Alejandro Samper
Keyword(s):  
Lower Bound ◽  
Convex Body ◽  
Hausdorff Distance ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Simplicial Polytopes ◽  
The Face ◽  
International Audience ◽  
Bound Theorem ◽  
Corps Convexe

International audience The face numbers of simplicial polytopes that approximate $C^1$-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence $\{P_n\}_{n=0}^{\infty}$ of simplicial polytopes converges to a $C^1$-convex body in the Hausdorff distance, then the entries of the $g$-vector of $P_n$ converge to infinity. Nous étudions les nombres de faces de polytopes simpliciaux qui se rapprochent de $C^1$-corps convexes dans la métrique Hausdorff. Plusieurs résultats structurels sur le skeleta de ces polytopes sont recherchées et utilisées pour calculer un théorème limite inférieure de cette classe de polytopes. Cela résout partiellement une conjecture formulée par Kalai en 1994: si une suite $\{P_n\}_{n=0}^{\infty}$ de polytopes simpliciaux converge vers une $C^1$-corps convexe dans la distance Hausdorff, puis les entrées du $g$-vecteur de $P_n$ convergent vers l’infini.

scholarly journals Inequalities for the Surface Area of Projections of Convex Bodies

2018 ◽  
Vol 70 (4) ◽  
pp. 804-823 ◽  
Author(s):  
Apostolos Giannopoulos ◽  
Alexander Koldobsky ◽  
Petros Valettas
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Convex Bodies ◽  
Maximal Surface ◽  
Projection Body ◽  
Projections Of Convex Bodies ◽  
Lower Dimensional

AbstractWe provide general inequalities that compare the surface area S(K) of a convex body K in ℝn to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

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.

A Note on Covering by Convex Bodies

2009 ◽  
Vol 52 (3) ◽  
pp. 361-365 ◽  
Author(s):  
Fejes Tóth Gábor
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Classical Theorem ◽  
Lattice Arrangement

AbstractA classical theorem of Rogers states that for any convex body K in n-dimensional Euclidean space there exists a covering of the space by translates of K with density not exceeding n log n + n log log n + 5n. Rogers’ theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of n the same bound can be attained by a covering which is the union of O(log n) translates of a lattice arrangement of K.

scholarly journals An elementary introduction to the geometry of quantum states with pictures

2019 ◽  
Vol 32 (02) ◽  
pp. 2030001 ◽  
Author(s):  
J. Avron ◽  
O. Kenneth
Keyword(s):  
Convex Body ◽  
Cross Sections ◽  
Convex Bodies ◽  
Quantum States ◽  
Entangled States ◽  
High Dimensions ◽  
Geometric Properties ◽  
Precise Sense ◽  
Entanglement Witnesses ◽  
Elementary Methods

This is a review of the geometry of quantum states using elementary methods and pictures. Quantum states are represented by a convex body, often in high dimensions. In the case of [Formula: see text] qubits, the dimension is exponentially large in [Formula: see text]. The space of states can be visualized, to some extent, by its simple cross sections: Regular simplexes, balls and hyper-octahedra. a When the dimension gets large, there is a precise sense in which the space of states resembles, almost in every direction, a ball. The ball turns out to be a ball of rather low purity states. We also address some of the corresponding, but harder, geometric properties of separable and entangled states and entanglement witnesses. “All convex bodies behave a bit like Euclidean balls.” Keith Ball

Continuity of the solution to the even Lp Minkowski problem for 0 < p < 1 in the plane

2020 ◽  
Vol 31 (12) ◽  
pp. 2050101
Author(s):  
Hejun Wang ◽  
Yusha Lv
Keyword(s):  
Weak Convergence ◽  
Surface Area ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Minkowski Problem

This paper concerns the continuity of the solution to the even [Formula: see text] Minkowski problem in the plane. When [Formula: see text], it is proved that the weak convergence of the even [Formula: see text] surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric. Moreover, the continuity of the solution to the even [Formula: see text] Minkowski problem with respect to [Formula: see text] is also obtained.

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