Random polytopes in a ball

1984 ◽  
Vol 21 (4) ◽  
pp. 753-762 ◽  
Author(s):  
C. Buchta ◽  
J. Müller
Keyword(s):  
Convex Hull ◽  
Surface Area ◽  
Convex Polytope ◽  
Expected Number ◽  
Random Polytopes ◽  
Mean Width ◽  
Dimensional Ball

The convex hull of n random points chosen independently and uniformly from a d-dimensional ball is a convex polytope. Its expected surface area, its expected mean width and its expected number of facets are explicitly determined.

Random polytopes in a ball

1984 ◽  
Vol 21 (04) ◽  
pp. 753-762 ◽  
Author(s):  
C. Buchta ◽  
J. Müller
Keyword(s):  
Convex Hull ◽  
Surface Area ◽  
Convex Polytope ◽  
Expected Number ◽  
Random Polytopes ◽  
Mean Width ◽  
Dimensional Ball

The convex hull of n random points chosen independently and uniformly from a d-dimensional ball is a convex polytope. Its expected surface area, its expected mean width and its expected number of facets are explicitly determined.

scholarly journals On the approximation of a ball by random polytopes

1994 ◽  
Vol 26 (4) ◽  
pp. 876-892 ◽  
Author(s):  
K.-H. Küfer
Keyword(s):  
Convex Hull ◽  
Unit Ball ◽  
Surface Area ◽  
Limiting Distribution ◽  
Spherically Symmetric ◽  
Random Vectors ◽  
Symmetric Distributions ◽  
Random Polytopes ◽  
Random Polytope ◽  
Dimensional Unit

Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a1,· ··, an. Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn)/Ε (Δ (Xn)) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets of Xn.

scholarly journals On the approximation of a ball by random polytopes

1994 ◽  
Vol 26 (04) ◽  
pp. 876-892 ◽  
Author(s):  
K.-H. Küfer
Keyword(s):  
Convex Hull ◽  
Unit Ball ◽  
Surface Area ◽  
Limiting Distribution ◽  
Spherically Symmetric ◽  
Random Vectors ◽  
Symmetric Distributions ◽  
Random Polytopes ◽  
Random Polytope ◽  
Dimensional Unit

Letbe a sequence of independent and identically distributed random vectors drawn from thed-dimensional unit ballBdand letXnbe the random polytope generated as the convex hull ofa1,· ··,an.Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributedaiand2 we prove that the limiting distribution of Δ(Xn)/Ε(Δ(Xn)) forn→ ∞ (satisfies a 0–1 law. In particular, we show that Varforn→ ∞. We provide analogous results for spherically symmetric distributions inBdwith regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets ofXn.

scholarly journals Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

2008 ◽  
Vol 60 (1) ◽  
pp. 3-32 ◽  
Author(s):  
Károly Böröczky ◽  
Károly J. Böröczky ◽  
Carsten Schütt ◽  
Gergely Wintsche
Keyword(s):  
Unit Ball ◽  
Surface Area ◽  
Convex Bodies ◽  
Extreme Points ◽  
Thin Shells ◽  
Minimal Volume ◽  
Asymptotic Formulae ◽  
Mean Width

AbstractGiven r > 1, we consider convex bodies in En which contain a fixed unit ball, and whose extreme points are of distance at least r from the centre of the unit ball, and we investigate how well these convex bodies approximate the unit ball in terms of volume, surface area and mean width. As r tends to one, we prove asymptotic formulae for the error of the approximation, and provide good estimates on the involved constants depending on the dimension.

scholarly journals Calculus rules for combinations of ellipsoids and applications

1993 ◽  
Vol 47 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alberto Seeger
Keyword(s):  
Convex Hull ◽  
Convex Polytope ◽  
Finite Family ◽  
Minkowski Sum ◽  
Calculus Rules ◽  
Ellipsoidal Approximations

We derive formulas for the Minkowski sum, the convex hull, the intersection, and the inverse sum of a finite family of ellipsoids. We show how these formulas can be used to obtain inner and outer ellipsoidal approximations of a convex polytope.

scholarly journals The maximum surface area polyhedron with five vertices inscribed in the sphere {\bb S}^{2}

2021 ◽  
Vol 77 (1) ◽  
pp. 67-74
Author(s):  
Jessica Donahue ◽  
Steven Hoehner ◽  
Ben Li
Keyword(s):  
Convex Hull ◽  
Surface Area ◽  
Coordination Polyhedron ◽  
Unit Sphere ◽  
The Other ◽  
Optimal Placement ◽  
The North ◽  
Trigonal Bipyramidal ◽  
Bipyramidal Structure ◽  
North And South

This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere {\bb S}^{2} so that the surface area of the convex hull of the points is maximized. It is shown that the optimal polyhedron has a trigonal bipyramidal structure with two vertices placed at the north and south poles and the other three vertices forming an equilateral triangle inscribed in the equator. This result confirms a conjecture of Akkiraju, who conducted a numerical search for the maximizer. As an application to crystallography, the surface area discrepancy is considered as a measure of distortion between an observed coordination polyhedron and an ideal one. The main result yields a formula for the surface area discrepancy of any coordination polyhedron with five vertices.

The convex hull of a uniform sample from the interior of a simple d-polytope

1989 ◽  
Vol 26 (02) ◽  
pp. 259-273 ◽  
Author(s):  
Barthold F. Van Wel
Keyword(s):  
Convex Hull ◽  
Asymptotic Expression ◽  
Expected Number ◽  
Simple Polytope ◽  
Uniform Sample ◽  
Sample Points

An asymptotic expression is given for the expected number of vertices of the convex hull of a uniform sample from the interior of a d-dimensional simple polytope. This extends a result derived by Rényi and Sulanke for sample points in the plane.

scholarly journals The Expected Number of Extreme Discs

2019 ◽  
Vol 35 (2) ◽  
Author(s):  
Nam-Dung Hoang ◽  
Nguyen Kieu Linh
Keyword(s):  
Computational Geometry ◽  
Convex Hull ◽  
Subject Classification ◽  
Expected Number ◽  
Finite Set ◽  
Expected Complexity

Abstract: Given a finite set D of n planar discs whose centers are distributed randomly. We are interested in the expected number of extreme discs of the convex hull of D. We show that the expected number of extreme discs is at most O(log2n) for any distribution. This result can be used to derive expected complexity of convex hull algorithms. Keywords: Convex hull, computational geometry, expected number. Mathematics Subject Classification (2010): 65D18, 52A15, 51N05.

scholarly journals Evaluation of Scrotal Temperature in Different Testicular Shapes by Infrared Thermography in Braford Bulls

2019 ◽  
Vol 47 (1) ◽  
Author(s):  
Patrícia Da Cruz Favaro ◽  
Gabriel Ribas Pereira ◽  
Flávio Antônio Barca Jr. ◽  
Marcelo Marcondes Seneda ◽  
Augusto César Alves Assunção ◽  
...  
Keyword(s):  
Surface Area ◽  
Infrared Thermography ◽  
Average Length ◽  
Thermal Sensitivity ◽  
The Body ◽  
Average Width ◽  
Scrotal Circumference ◽  
Mean Width ◽  
The Mean ◽  
Oval Shape

Background: The mechanisms of testicular thermoregulation may influence the blood flow provided by the testicular artery, where the proper blood supply to the testicle is crucial for promotingspermatogenesis and reproductive function in bulls. The size and shape of the testicles are determined by genetic mechanisms and environmental effects. A better understanding of the relationships between the anatomical characteristics of the testicles and scrotum can support a better reproductive assessment. The purpose of the current study was to evaluate the testicular temperature of different scrotal shapes using infrared thermography in bulls.Materials, Methods & Results: We evaluated 132 Braford bulls with an average age of 24 months. The evaluation of breeding bull semen was performed prior to the beginning of the experiment. Then, animals were selected on the basis of the size of their testes, which was determined by dividing the average width by the average length. The scrotal circumference was measured with a millimeter tape positioned around the largest circumference. Testicular and ocular temperature measurements and analysis were conducted using an infrared thermal camera, Flir T440 with emissivity of 0.98 and thermal sensitivity of 0.05°C. Testicular scrotum temperature and testicular shape were analyzed with one-way ANOVA using Minitab 16, and values of P < 0.05 were considered statistically significant. We observed that 67.42% of testicle shapes were long-oval, and 32.58% were long-moderate. The testicular temperature was higher in bulls with the long-moderate shape compared to those with the long-oval shape (P < 0.05). The mean length was higher in long-moderate shaped testicles compared to those of the long-oval shape (P < 0.01). There was no significant differences in rectal and ocular temperatures or in scrotal circumference between bulls with long-moderate and long-oval shapes (P > 0.05). In addition, the mean width was lower in testicles of long-moderate shape compared to those of the long-oval scrotal format (P < 0.01).Discussion: The results obtained showed that Braford bulls with the long-moderate testicular shape have a higher testicular temperature to maintain proper thermoregulation. The present study demonstrated that IRT can be used to evaluate the testicular temperature in animals with different scrotal conformations. In this study, Braford bulls showed lower length and width values for animals having long-moderate (9.21 and 5.22, respectively) and long-oval formats (8.56 and 5.56, respectively). In contrast, previous reports examining Nellore bulls between the ages of 17-20 months found a predominance of the long oval shape followed by the long-moderate shape, which indicates a change in testicular shape as age progresses, resulting in a rounder testicular shape. Perhaps other factors, such as the external cremaster muscle and tunica dartos, cause the testicles to be retracted towards the body at lower temperatures while at high temperatures, relaxation occurs. The prevailing testicular shape in Braford animals with a mean age of 24 months was the long-moderate shape. Thus, testicles with a larger surface area will have lower temperatures because they can dissipate heat more easily than testicles with lower surface area. The results suggest that the long-moderate scrotum format may influence the testicular temperature in mature Braford bulls.

Comparing the M-position with some classical positions of convex bodies

2011 ◽  
Vol 152 (1) ◽  
pp. 131-152 ◽  
Author(s):  
E. MARKESSINIS ◽  
G. PAOURIS ◽  
CH. SAROGLOU
Keyword(s):  
Minimal Surface ◽  
Surface Area ◽  
Convex Bodies ◽  
Mean Width ◽  
Quantitative Results ◽  
Minimal Surface Area

AbstractThe purpose of this paper is to compare some classical positions of convex bodies. We provide exact quantitative results which show that the minimal surface area position and the minimal mean width position are not necessarily M-positions. We also construct examples of unconditional convex bodies of minimal surface area that exhibit the worst possible behavior with respect to their mean width or their minimal hyperplane projection.

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