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.

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