Intrinsic volumes of inscribed random polytopes in smooth convex bodies

LetKbe ad-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote byKnthe convex hull ofnpoints chosen randomly and independently fromKaccording to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of thesth intrinsic volumesVs(Kn) ofKnfors∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes ofKn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

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Further inequalities for the (generalized) Wills functional

Communications in Contemporary Mathematics ◽

10.1142/s021919972050011x ◽

2020 ◽

pp. 2050011

Author(s):

David Alonso-Gutiérrez ◽

María A. Hernández Cifre ◽

Jesús Yepes Nicolás

Keyword(s):

Convex Body ◽

Convex Bodies ◽

General Setting ◽

Concave Function ◽

Upper Bounds ◽

Edge Length ◽

Symmetric Convex ◽

Concave Functions ◽

Intrinsic Volumes

The Wills functional [Formula: see text] of a convex body [Formula: see text], defined as the sum of its intrinsic volumes [Formula: see text], turns out to have many interesting applications and properties. In this paper, we make profit of the fact that it can be represented as the integral of a log-concave function, which, furthermore, is the Asplund product of other two log-concave functions, and obtain new properties of the Wills functional (indeed, we will work in a more general setting). Among others, we get upper bounds for [Formula: see text] in terms of the volume of [Formula: see text], as well as Brunn–Minkowski and Rogers–Shephard-type inequalities for this functional. We also show that the cube of edge-length 2 maximizes [Formula: see text] among all [Formula: see text]-symmetric convex bodies in John position, and we reprove the well-known McMullen’s inequality [Formula: see text] using a different approach.

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Weighted faces of Poisson hyperplane tessellations

Advances in Applied Probability ◽

10.1017/s0001867800003529 ◽

2009 ◽

Vol 41 (03) ◽

pp. 682-694 ◽

Cited By ~ 2

Author(s):

Rolf Schneider

Keyword(s):

Euclidean Space ◽

Upper Bounds ◽

Dimensional Euclidean Space ◽

Lower And Upper Bounds ◽

Intrinsic Volumes ◽

Vertex Number ◽

Zero Cell ◽

Lower Dimensional

We study lower-dimensional volume-weighted typical faces of a stationary Poisson hyperplane tessellation in d-dimensional Euclidean space. After showing how their distribution can be derived from that of the zero cell, we obtain sharp lower and upper bounds for the expected vertex number of the volume-weighted typical k-face (k=2,…,d). The bounds are respectively attained by parallel mosaics and by isotropic tessellations. We conclude with a remark on expected face numbers and expected intrinsic volumes of the zero cell.

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The Characteristic Properties of the Minimal Lp-Mean Width

Journal of Function Spaces ◽

10.1155/2017/2943073 ◽

2017 ◽

Vol 2017 ◽

pp. 1-10

Author(s):

Tongyi Ma

Keyword(s):

Convex Body ◽

Existence And Uniqueness ◽

Convex Bodies ◽

Smooth Convex ◽

Smooth Convex Body ◽

Minimal Position ◽

Mean Width ◽

Suitable Measure ◽

Minimum Position

Giannopoulos proved that a smooth convex body K has minimal mean width position if and only if the measure hK(u)σ(du), supported on Sn-1, is isotropic. Further, Yuan and Leng extended the minimal mean width to the minimal Lp-mean width and characterized the minimal position of convex bodies in terms of isotropicity of a suitable measure. In this paper, we study the minimal Lp-mean width of convex bodies and prove the existence and uniqueness of the minimal Lp-mean width in its SL(n) images. In addition, we establish a characterization of the minimal Lp-mean width, conclude the average Mp(K) with a variation of the minimal Lp-mean width position, and give the condition for the minimum position of Mp(K).

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On the geodetic hull number for complementary prisms II

RAIRO - Operations Research ◽

10.1051/ro/2020089 ◽

2020 ◽

Author(s):

Diane Castonguay ◽

Erika Morais Martins Coelho ◽

Hebert Coelho ◽

Julliano Nascimento

Keyword(s):

Convex Hull ◽

Shortest Path ◽

Disjoint Union ◽

Perfect Matching ◽

Convex Set ◽

Upper Bounds ◽

The Other ◽

Lower And Upper Bounds ◽

Split Graph ◽

Complementary Prism

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is \textit{convex} if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The \textit{convex hull} $H(S)$ of $S$ is the smallest convex set containing $S$. If $H(S) = V(G)$, then $S$ is a \textit{hull set}. The cardinality $h(G)$ of a minimum hull set of $G$ is the \textit{hull number} of $G$. The \textit{complementary prism} $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. A graph $G$ is \textit{autoconnected} if both $G$ and $\overline{G}$ are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When $G$ is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when $G$ is a non-split graph, it is limited by $3$.

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A SPHERICAL MAPPING AND BORSUK CONJECTURE IN RIEMANNIAN AND NON-EUCLIDEAN SPACES

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.25.1994.4436 ◽

1994 ◽

Vol 25 (2) ◽

pp. 149-155

Author(s):

BORIS V. DEKSTER

Keyword(s):

Convex Body ◽

Riemannian Manifolds ◽

Additional Assumption ◽

Convex Bodies ◽

Smooth Convex ◽

Euclidean Spaces ◽

Smooth Convex Body ◽

Bounded Set

We introduce an analog of the spherical mapping for convex bodies in a Riemannian $n$-manifold, and then use this construction to prove the Borsuk conjecture for some types of such bodies. The Borsuk conjecture is that each bounded set $X$ in the Euclidean $n$-space can be covered by $n +1$ sets of smaller diameter. The conjecture was disproved recently by Kahn and Kalai. However Hadwiger proved the Borsuk conjecture under the additional assumption that the set $X$ is a smooth convex body. Here we extend this result to convex bodies in Riemannian manifolds under some further restrictions.

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Weighted faces of Poisson hyperplane tessellations

Advances in Applied Probability ◽

10.1239/aap/1253281060 ◽

2009 ◽

Vol 41 (3) ◽

pp. 682-694 ◽

Cited By ~ 12

Author(s):

Rolf Schneider

Keyword(s):

Euclidean Space ◽

Upper Bounds ◽

Dimensional Euclidean Space ◽

Lower And Upper Bounds ◽

Intrinsic Volumes ◽

Vertex Number ◽

Zero Cell ◽

Lower Dimensional

We study lower-dimensional volume-weighted typical faces of a stationary Poisson hyperplane tessellation in d-dimensional Euclidean space. After showing how their distribution can be derived from that of the zero cell, we obtain sharp lower and upper bounds for the expected vertex number of the volume-weighted typical k-face (k=2,…,d). The bounds are respectively attained by parallel mosaics and by isotropic tessellations. We conclude with a remark on expected face numbers and expected intrinsic volumes of the zero cell.

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Online estimation of lower and upper bounds for heart sound boundaries in chest sound using Convex-hull algorithm

2012 Annual International Conference of the IEEE Engineering in Medicine and Biology Society ◽

10.1109/embc.2012.6346906 ◽

2012 ◽

Author(s):

F. Caglar ◽

I. Y. Ozbek

Keyword(s):

Convex Hull ◽

Heart Sound ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Online Estimation ◽

Convex Hull Algorithm

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Bounds for Covering Symmetric Convex Bodies by a Lattice Congruent to a Given Lattice

Journal of Mathematics Research ◽

10.5539/jmr.v9n2p84 ◽

2017 ◽

Vol 9 (2) ◽

pp. 84

Author(s):

Beomjong Kwak

Keyword(s):

Convex Body ◽

Convex Bodies ◽

Upper Bounds ◽

Symmetric Convex Body ◽

Symmetric Convex ◽

Lattice Covering ◽

Centrally Symmetric

In this paper, we focus on lattice covering of centrally symmetric convex body on $\mathbb{R}^2$. While there is no constraint on the lattice in many other results about lattice covering, in this study, we only consider lattices congruent to a given lattice to retain more information on the lattice. To obtain some upper bounds on the infimum of the density of such covering, we will say a body is a coverable body with respect to a lattice if such lattice covering is possible, and try to suggest a function of a given lattice such that any centrally symmetric convex body whose area is not less than the function is a coverable body. As an application of this result, we will suggest a theorem which enables one to apply this to a coverable body to suggesting an efficient lattice covering for general sets, which may be non-convex and may have holes.

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A Note on the Borsuk Conjecture

Canadian Mathematical Bulletin ◽

10.4153/cmb-1967-001-1 ◽

1967 ◽

Vol 10 (1) ◽

pp. 1-3

Author(s):

Z.A. Melzak

Keyword(s):

Convex Body ◽

Constant Width ◽

Convex Bodies ◽

Bounded Subset ◽

Smooth Convex ◽

Support Plane ◽

Smooth Convex Body

According to the still unproved conjecture of Borsuk [1] a bounded subset A of the Euclidean n-space En is a union of n + 1 sets of diameters less than the diameter D of A. Since A can be imbedded in a set of constant width D, [2], it may be assumed that A is already of constant width. If in addition A is smooth, i. e., if through every point of its boundary ∂A there passes one and only one support plane of A, then the truth of Borsuk′s conjecture can be proved very easily [3]. The question arises whether Borsuk′s conjecture holds also for arbitrary smooth convex bodies, not merely for those of constant width. Since it is not known whether a smooth convex body K can be imbedded in a smooth set of constant width D, the answer is not immediate. In this note we show that the answer is affirmative.

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