1. CMB 2011 (vol 55 pp. 498)
2. CMB 2009 (vol 52 pp. 342)
 Bezdek, K.; Kiss, Gy.

On the Xray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width
The Xray numbers of some classes of convex bodies are investigated.
In particular, we give a proof of the Xray 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, Xray number, Xray 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 RogersShephard 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 crosspolytope. 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 BanachMazur distance to the
regular hexagon.
Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, BanachMazur compactum, (modified) BanachMazur distance, volume ratio Categories:52A40, 46B20 
