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.

scholarly journals Duality theorem for approximation problem with fuzzy statement

2015 ◽  
Vol 23 ◽  
pp. 88
Author(s):  
V.I. Ruban
Keyword(s):  
Best Approximation ◽  
Duality Theorem ◽  
Convex Set ◽  
Approximation Problem ◽  
The Best Approximation

We obtained fuzzy analogue of duality theorem for the best approximation by a convex set.

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} .

On Measures of Symmetry of Convex Bodies

1965 ◽  
Vol 17 ◽  
pp. 497-504 ◽  
Author(s):  
G. D. Chakerian ◽  
S. K. Stein
Keyword(s):  
Convex Body ◽  
Convex Set ◽  
Compact Convex ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Maximum Volume ◽  
Compact Convex Set ◽  
Centrally Symmetric ◽  
Measures Of Symmetry ◽  
Interior Points

Let K be a convex body (compact, convex set with interior points) in n-dimensional Euclidean space En, and let V(K) denote the volume of K. Let K′ be a centrally symmetric body of maximum volume contained in K (in fact, K′ is unique; see 2 or 9), and definec(K) = V(K′)/V(K)Letc(n) = inf{c(K) : K ⊂ En}.

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.

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)).

On a random variable related to a system of convex bodies in the Euclidean space En

1998 ◽  
Vol 30 (2) ◽  
pp. 335-341
Author(s):  
Marius Stoka
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Convex Set ◽  
Convex Bodies ◽  
Random Variable ◽  
Second Moment

Let us consider, in the Euclidean space En, a fixed n-dimensional convex body K0 of volume V0 and a system K1,…,Km of mn-dimensional convex bodies, congruent to a convex set K. Assume that the sets Ki (i = 1,…,m) have random positions, being stochastically independent and uniformly distributed on a limited domain of En and denote by Vm the volume of the convex body Km = K0 ∩ (K1 ∩ … ∩ Km). The aim of this paper is the evaluation of the second moment of the random variable Vm.

On a random variable related to a system of convex bodies in the Euclidean space E n

1998 ◽  
Vol 30 (02) ◽  
pp. 335-341
Author(s):  
Marius Stoka
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Convex Set ◽  
Convex Bodies ◽  
Random Variable ◽  
Second Moment

Let us consider, in the Euclidean space E n , a fixed n-dimensional convex body K 0 of volume V 0 and a system K 1,…,K m of m n-dimensional convex bodies, congruent to a convex set K. Assume that the sets K i (i = 1,…,m) have random positions, being stochastically independent and uniformly distributed on a limited domain of E n and denote by V m the volume of the convex body K m = K 0 ∩ (K 1 ∩ … ∩ K m ). The aim of this paper is the evaluation of the second moment of the random variable V m .

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 Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes

2019 ◽  
Author(s):  
Florian Besau ◽  
Steven Hoehner ◽  
Gil Kur
Keyword(s):  
Unit Ball ◽  
Best Approximation ◽  
Convex Bodies ◽  
Asymptotic Formulas ◽  
Smooth Convex ◽  
High Dimensions ◽  
Sharp Bounds ◽  
Intrinsic Volumes ◽  
The Best Approximation ◽  
Euclidean Unit Ball

Abstract We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope that is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants. Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak’s dual Brunn–Minkowski theory.

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