Asymptotics of the diffraction field on a smooth convex body in the neighborhood of the edge of a nonsingular caustic

1991 ◽  
Vol 55 (3) ◽  
pp. 1645-1656
Author(s):  
I. V. Andronov
Keyword(s):  
Convex Body ◽  
Diffraction Field ◽  
Smooth Convex ◽  
Smooth Convex Body

Diffraction of a nonregular short wave by a smooth convex body in the penumbra

1984 ◽  
Vol 24 (3) ◽  
pp. 274-282
Author(s):  
I. Ya. Bilyi
Keyword(s):  
Convex Body ◽  
Short Wave ◽  
Smooth Convex ◽  
Smooth Convex Body

How Smooth is the Shadow of a Smooth Convex Body?

1986 ◽  
Vol s2-33 (1) ◽  
pp. 101-109 ◽  
Author(s):  
Christer O. Kiselman
Keyword(s):  
Convex Body ◽  
Smooth Convex ◽  
Smooth Convex Body

scholarly journals The Characteristic Properties of the Minimal Lp-Mean Width

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).

Shortwave Diffraction from a Smooth Convex Body

1979 ◽  
pp. 83-116
Author(s):  
Vasilii M. Babič ◽  
Natal’ya Y. Kirpičnikova
Keyword(s):  
Convex Body ◽  
Smooth Convex ◽  
Smooth Convex Body

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

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.

scholarly journals Mean width of random polytopes in a reasonably smooth convex body

2009 ◽  
Vol 100 (10) ◽  
pp. 2287-2295 ◽  
Author(s):  
K.J. Böröczky ◽  
F. Fodor ◽  
M. Reitzner ◽  
V. Vígh
Keyword(s):  
Convex Body ◽  
Smooth Convex ◽  
Random Polytopes ◽  
Smooth Convex Body ◽  
Mean Width

A Note on the Borsuk Conjecture

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.

scholarly journals Intrinsic volumes of inscribed random polytopes in smooth convex bodies

2010 ◽  
Vol 42 (3) ◽  
pp. 605-619 ◽  
Author(s):  
I. Bárány ◽  
F. Fodor ◽  
V. Vígh
Keyword(s):  
Convex Body ◽  
Convex Hull ◽  
Convex Bodies ◽  
Upper Bounds ◽  
Smooth Convex ◽  
Lower And Upper Bounds ◽  
Covering Theorem ◽  
Intrinsic Volumes ◽  
Large Numbers ◽  
Distribution Matching

Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by Kn the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes Vs(Kn) of Kn for s ∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of Kn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

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