On the illumination of smooth convex bodies

1992 ◽  
Vol 58 (6) ◽  
pp. 611-614 ◽  
Author(s):  
K�roly Bezdek
Keyword(s):  
Convex Bodies ◽  
Smooth Convex

Source radiation in the presence of smooth convex bodies

Radio Science ◽  
1979 ◽  
Vol 14 (2) ◽  
pp. 217-237 ◽  
Author(s):  
R. Mittra ◽  
S. Safavi-Naini
Keyword(s):  
Convex Bodies ◽  
Smooth Convex ◽  
Source Radiation

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).

scholarly journals Smooth convex bodies with proportional projection functions

2007 ◽  
Vol 159 (1) ◽  
pp. 317-341 ◽  
Author(s):  
Ralph Howard ◽  
Daniel Hug
Keyword(s):  
Convex Bodies ◽  
Smooth Convex ◽  
Projection Functions

Volume approximation of smooth convex bodies by three-polytopes of restricted number of edges

2007 ◽  
Vol 153 (1) ◽  
pp. 25-48 ◽  
Author(s):  
Károly J. Böröczky ◽  
Salvador S. Gomis ◽  
Péter Tick
Keyword(s):  
Convex Bodies ◽  
Smooth Convex ◽  
Restricted Number ◽  
Volume Approximation

scholarly journals Approximation of General Smooth Convex Bodies

2000 ◽  
Vol 153 (2) ◽  
pp. 325-341 ◽  
Author(s):  
Károly Böröczky
Keyword(s):  
Convex Bodies ◽  
Smooth Convex

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.

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