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Search: All articles in the CMB digital archive with keyword convex body

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1. CMB 2011 (vol 55 pp. 498)

Fradelizi, Matthieu; Paouris, Grigoris; Schütt, Carsten
Simplices in the Euclidean Ball
We establish some inequalities for the second moment $$ \frac{1}{|K|} \int_{K}|x|_2^2 \,dx $$ of a convex body $K$ under various assumptions on the position of $K$.

Keywords:convex body, simplex

2. CMB 2009 (vol 52 pp. 342)

Bezdek, K.; Kiss, Gy.
On the X-ray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width
The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions $3$, $4$, $5$ and $6$.

Keywords:almost smooth convex body, convex body of constant width, weakly neighbourly antipodal convex polytope, Illumination Conjecture, X-ray number, X-ray Conjecture
Categories:52A20, 52A37, 52C17, 52C35

3. CMB 2006 (vol 49 pp. 185)

Averkov, Gennadiy
On the Inequality for Volume and Minkowskian Thickness
Given a centrally symmetric convex body $B$ in $\E^d,$ we denote by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex body in $\M^d(B).$ The relationship between volume $V(K)$ and the Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can naturally be given by the sharp geometric inequality $V(K) \ge \alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple corollary of the Rogers--Shephard inequality we obtain that $\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach--Mazur distance to the regular hexagon.

Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, Banach-Mazur compactum, (modified) Banach-Mazur distance, volume ratio
Categories:52A40, 46B20

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