Expand all Collapse all | Results 1 - 6 of 6 |
1. CMB 2011 (vol 55 pp. 498)
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 Category:52A20 |
2. CMB 2009 (vol 52 pp. 342)
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 2009 (vol 52 pp. 380)
Successive Minima and Radii In this note we present inequalities relating the successive minima of an
$o$-symmetric convex body and the successive inner and outer radii of the
body. These inequalities join known inequalities involving only either
the successive minima or the successive radii.
Keywords:successive minima, inner and outer radii Categories:52A20, 52C07, 52A40, 52A39 |
4. CMB 2009 (vol 52 pp. 464)
Two Volume Product Inequalities and Their Applications Let $K \subset {\mathbb{R}}^{n+1}$ be a convex body of class $C^2$
with everywhere positive Gauss curvature. We show that there exists
a positive number $\delta (K)$ such that for any $\delta \in (0,
\delta(K))$ we have $\Volu(K_{\delta})\cdot
\Volu((K_{\delta})^{\sstar}) \geq \Volu(K)\cdot \Volu(K^{\sstar}) \geq
\Volu(K^{\delta})\cdot \Volu((K^{\delta})^{\sstar})$, where $K_{\delta}$,
$K^{\delta}$ and $K^{\sstar}$ stand for the convex floating body, the
illumination body, and the polar of $K$, respectively. We derive a
few consequences of these inequalities.
Keywords:affine invariants, convex floating bodies, illumination bodies Categories:52A40, 52A38, 52A20 |
5. CMB 2004 (vol 47 pp. 246)
On Maximal $k$-Sections and Related Common Transversals of Convex Bodies Generalizing results from [MM1] referring
to the intersection body $IK$ and
the cross-section body $CK$ of a convex body
$K \subset \sR^d, \, d \ge 2$,
we prove theorems about maximal $k$-sections of convex bodies,
$k \in \{1, \dots, d-1\}$,
and, simultaneously, statements
about common maximal
$(d-1)$- and $1$-transversals of families
of convex bodies.
Categories:52A20, 55Mxx |
6. CMB 2002 (vol 45 pp. 232)
On Strongly Convex Indicatrices in Minkowski Geometry The geometry of indicatrices is the foundation of Minkowski geometry.
A strongly convex indicatrix in a vector space is a strongly convex
hypersurface. It admits a Riemannian metric and has a distinguished
invariant---(Cartan) torsion. We prove the existence of non-trivial
strongly convex indicatrices with vanishing mean torsion and discuss
the relationship between the mean torsion and the Riemannian curvature
tensor for indicatrices of Randers type.
Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35 |