Isoperimetric inequalities for the mean width of a convex body

1973 ◽  
Vol 1 (3) ◽  
Author(s):  
G.D. Chakerian
Keyword(s):  
Convex Body ◽  
Isoperimetric Inequalities ◽  
Mean Width ◽  
The Mean

scholarly journals The Mean Width of Circumscribed Random Polytopes

2010 ◽  
Vol 53 (4) ◽  
pp. 614-628 ◽  
Author(s):  
Károly J. Böröczky ◽  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Asymptotic Formula ◽  
Uniform Distribution ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Simplicial Polytope ◽  
Mean Width ◽  
The Mean ◽  
The Difference ◽  
Precise Asymptotic

AbstractFor a given convex body K in ℝd, a random polytope K(n) is defined (essentially) as the intersection of n independent closed halfspaces containing K and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of K(n) and K as n tends to infinity. For a simplicial polytope P, a precise asymptotic formula for the difference of the mean widths of P(n) and P is obtained.

scholarly journals Interaction of Poisson hyperplane processes and convex bodies

2019 ◽  
Vol 56 (4) ◽  
pp. 1020-1032 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Random Polytope ◽  
Mean Width ◽  
The Mean

AbstractGiven a stationary and isotropic Poisson hyperplane process and a convex body K in ${\mathbb R}^d$ , we consider the random polytope defined by the intersection of all closed half-spaces containing K that are bounded by hyperplanes of the process not intersecting K. We investigate how well the expected mean width of this random polytope approximates the mean width of K if the intensity of the hyperplane process tends to infinity.

On the volume of caps and bounding the mean-width of an isotropic convex body

2010 ◽  
Vol 149 (2) ◽  
pp. 317-331 ◽  
Author(s):  
PETER PIVOVAROV
Keyword(s):  
Convex Body ◽  
Lower Bounds ◽  
Tail Behavior ◽  
Linear Functionals ◽  
Converse Statement ◽  
Isotropic Position ◽  
Mean Width ◽  
The Mean

AbstractLetKbe a convex body which is (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting onKexhibit super-Gaussian tail behavior. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if anarbitraryisotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound its mean-width.

scholarly journals The mean width of random polytopes circumscribed around a convex body

2010 ◽  
Vol 81 (2) ◽  
pp. 499-523 ◽  
Author(s):  
Károly J. Böröczky ◽  
Ferenc Fodor ◽  
Daniel Hug
Keyword(s):  
Convex Body ◽  
Random Polytopes ◽  
Mean Width ◽  
The Mean

scholarly journals Ethnic variation in medial orbital wall anatomy and its implications for decompression surgery

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Minhui Amy Chan ◽  
Farah Ibrahim ◽  
Arjunan Kumaran ◽  
Kailing Yong ◽  
Anita Sook Yee Chan ◽  
...  
Keyword(s):  
Relative Position ◽  
Ethmoid Sinus ◽  
Decompression Surgery ◽  
Ethnic Variation ◽  
Sinus Wall ◽  
Orbital Wall ◽  
Mean Width ◽  
Significant Difference ◽  
The Mean ◽  
Medial Orbital Wall

Abstract Background To describe the inter-ethnic variation in medial orbital wall anatomy between Chinese, Malay, Indian and Caucasian subjects. Methods Single-centre, retrospective, Computed Tomography (CT)-based observational study. 20 subjects of each ethnicity, were matched for gender and laterality. We excluded subjects younger than 16 years and those with orbital pathology. OsiriX version 8.5.1 (Pixmeo., Switzerland) and DICOM image viewing software CARESTREAM Vue PACS (Carestream Health Inc., USA) were used to measure the ethmoidal sinus length, width and volume, medial orbital wall and floor angle and the relative position of the posterior ethmoid sinus to the posterior maxillary wall. Statistical analyses were performed using Statistical Package for Social Sciences version 25.0 (IBM, USA). Results There were 12 males (60 %) in each group, with no significant difference in age (p = 0.334–0.994). The mean ethmoid sinus length in Chinese, Malay, Indian and Caucasian subjects, using the Chinese as reference, were 37.2, 36.9, 38.0 and 37.4mm, the mean width was 11.6, 10.5, 11.4 and 10.0mm (p = 0.020) and the mean ethmoid sinus volume were 3362, 3652, 3349 and 3898mm3 respectively. The mean medial orbital wall and floor angle was 135.0, 131.4, 131.0 and 136.8 degrees and the mean relative position of posterior ethmoid sinus to posterior maxillary wall were − 2.0, -0.2, -1.5 and 1.6mm (p = 0.003) respectively. Conclusions No inter-ethnic variation was found in decompressible ethmoid sinus volume. Caucasians had their posterior maxillary sinus wall anterior to their posterior ethmoidal walls unlike the Chinese, Malay and Indians. Awareness of ethnic variation is essential for safe orbital decompression.

Cervical and High Thoracic Ligamentum Flavum Frequently Fails to Fuse in the Midline

2003 ◽  
Vol 99 (6) ◽  
pp. 1387-1390 ◽  
Author(s):  
Philipp Lirk ◽  
Christian Kolbitsch ◽  
Guenther Putz ◽  
Joshua Colvin ◽  
Hans Peter Colvin ◽  
...  
Keyword(s):  
Ligamentum Flavum ◽  
Epidural Needle ◽  
Complete Fusion ◽  
Thoracic Epidural ◽  
Loss Of Resistance ◽  
Ligamenta Flava ◽  
Mean Width ◽  
Human Cadavers ◽  
The Mean ◽  
High Thoracic Epidural

Background Cervical and high thoracic epidural anesthesia and analgesia have gained increasing importance in the treatment of painful conditions and as components of anesthetics for cardiac and breast surgery. In contrast to the hanging-drop technique, the loss-of-resistance technique is thought to rely on the penetration of the ligamentum flavum. However, the exact morphology of the ligamentum flavum at different vertebral levels remains controversial. Therefore, the aim of this study was to investigate the incidence and morphology of cervical and high thoracic ligamentum flavum mid-line gaps in embalmed cadavers. Methods Vertebral column specimens were obtained from 52 human cadavers. On each dissected level, ligamentum flavum mid-line gaps were recorded and evaluated with respect to shape and size. Results The following variations were encountered: complete fusion in the mid-line, mid-line fusion with a gap in the caudal part, mid-line gap, and mid-line gap with widened caudal end. The incidence of mid-line gaps at the following levels was: C3-C4: 66%, C4-C5: 58%, C5-C6: 74%, C6-C7: 64%, C7-T1: 51%, Th1-Th2: 21%, Th2-Th3: 11%, Th3-Th4: 4%, Th4-Th5: 2%, and Th5-Th6: 2%. The mean width of mid-line gaps was 1.0 +/- 0.3 mm. Conclusions In conclusion, the present study shows that gaps in the ligamenta flava are frequent at cervical and high thoracic levels but become rare at the T3/T4 level and below, such that one cannot always rely on the ligamentum flavum as a perceptible barrier to epidural needle placement at these levels.

scholarly journals Radiographic Measurement of Cochlear in Sudanese Population using High Resolution Computed Tomography (HRCT)

2021 ◽  
Vol 29 (2) ◽  
Author(s):  
Lubna Bushara ◽  
Mohamed Yousef ◽  
Ikhlas Abdelaziz ◽  
Mogahid Zidan ◽  
Dalia Bilal ◽  
...  
Keyword(s):  
Computed Tomography ◽  
Hearing Loss ◽  
High Resolution ◽  
Temporal Bone ◽  
Healthy Subjects ◽  
Cochlear Nerve ◽  
High Resolution Computed Tomography ◽  
Mean Width ◽  
The Mean ◽  
The Right

This study aimed to determine the measurements of the cochlea among healthy subjects and hearing deafness subjects using a High Resolution Computed Tomography (HRCT). A total of 230 temporal bone HRCT cases were retrospectively investigated in the period spanning from 2011 to 2015. Three 64-slice units were used to examine patients with clinical complaints of hearing loss conditions at three Radiology departments in Khartoum, Sudan. For the control group (A) healthy subjects, the mean width of the right and left cochlear were 5.61±0.40 mm and 5.56±0.58 mm, the height were 3.56±0.36 mm and 3.54±0.36 mm, the basal turn width were 1.87±0.19 mm and 1.88 ±0.18 mm, the width of the cochlear nerve canal were 2.02±1.23 and 1.93±0.20, cochlear nerve density was 279.41±159.02 and 306.84±336.9 HU respectively. However, for the experimental group (B), the mean width of the right and left cochlear width were 5.38±0.46 mm and 5.34±0.30 mm, the height were 3.53±0.25 mm and 3.49±0.28mm, the basal turn width were 1.76±0.13 mm, and 1.79±0.13 mm, the width of the cochlear nerve canal were 1.75±0.18mm and 1.73±0.18mm, and cochlear nerve density were 232.84±316.82 and 196.58±230.05 HU, respectively. The study found there was a significant difference in cochlea’s measurement between the two groups with a p-value < 0.05. This study had established baseline measurements for the cochlear for the healthy Sudanese population. Furthermore, it found that HRCT of the temporal bone was the best for investigation of the cochlear and could provide a guide for the clinicians to manage congenital hearing loss.

scholarly journals The Micro-Geometry Evaluation on Steel Sheets after Chemical Surface Treatment

2013 ◽  
Vol 10 (2) ◽  
pp. 22-25
Author(s):  
Dagmar Draganovská ◽  
Miroslav Tomáš ◽  
Peter Ižol
Keyword(s):  
Surface Treatment ◽  
Steel Sheet ◽  
Unit Length ◽  
Mean Deviation ◽  
Maximum Height ◽  
Chemical Surface ◽  
Steel Sheets ◽  
Mean Width ◽  
The Mean ◽  
Metal Materials

Abstract The protection of metal materials by coatings is one of the most important way to protect the surface against corrosion. Galvanisation of metal surfaces by an electric current is one of the most frequent way of surface treatment due to its wide application. The paper is oriented to the evaluation of the surface quality in terms of micro-geometry. The basic surface of the steel sheet S235JRG2 has been evaluted and compared to the degreased one, galvanized one and surface after galvanizing, chromatizing and phosphating. The microgeometry parameters such as the arithmetical mean deviation Ra, the maximum height of profile Rz, the total height of profile Rt as well as the the mean width of the profile elements RSm and the number of peaks per unit length RPc has been compared.

scholarly journals ISOPERIMETRIC INEQUALITIES FOR Lp GEOMINIMAL SURFACE AREA

2011 ◽  
Vol 53 (3) ◽  
pp. 717-726 ◽  
Author(s):  
BAOCHENG ZHU ◽  
NI LI ◽  
JIAZU ZHOU
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Type Inequality ◽  
Isoperimetric Inequalities ◽  
Surface Areas ◽  
Cyclic Inequality ◽  
Geominimal Surface Area

AbstractIn this paper, we establish a number of Lp-affine isoperimetric inequalities for Lp-geominimal surface area. In particular, we obtain a Blaschke–Santaló type inequality and a cyclic inequality between different Lp-geominimal surface areas of a convex body.

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