Let K be a convex body in ℝ 2 . We consider the Voronoi tessellation generated by homogeneous Poisson point process of intensity λ conditional on existence cell which contains K. When λ→∞, this converges from above to and we provide precise asymptotics expectation its defect area, perimeter number vertices. As Rényi Sulanke’s seminal papers random hulls, regularity has crucial importance deal w...

If a convex body K in R is contained in a convex body L of elliptic type (a curvature image), then it is known that the affine surface area of K is not larger than the affine surface area of L. We prove that the affine surface areas of K and L can only be equal if K = L. 2010 Mathematics Subject Classification: primary 52A10; secondary 53A15

The Chvátal-Gomory closure and the split closure of a rational polyhedron are rational polyhedra. It was recently shown that the Chvátal-Gomory closure of a strictly convex body is also a rational polytope. In this note, we show that the split closure of a strictly convex body is defined by a finite number of split disjunctions, but is not necessarily polyhedral. We also give a closed form expr...

We prove that if a convex body has absolutely continuous surface area measure, whose density is sufficiently close to the constant, then the sequence {ΠK} of convex bodies converges to the ball with respect to the Banach-Mazur distance, asm→∞. Here, Π denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty ...

In the present research, free convection heat transfer from isothermal concave and convex body shapes is studied numerically. The body shapes investigated here, are bi-sphere, cylinder, prolate and cylinder with hemispherical ends; besides, they have the same height over width (H/D = 2). A Numerical simulation is implemented to obtain heat transfer and fluid flow from all of the geometries in a...

Using the existence theorem of Minkowski, we consider the mapping of a data set in IR d into a convex body called the Minkowski polytope. Elsewhere the rst author has treated this as a data analytic tool. Here we show that the MP obeys a strong law in the sense of converging in the Hausdorr metric with increasing sample size to a convex body associated with the population distribution.

Let K<= R be a convex body and choose points xl,x2 xn randomly, independently, and uniformly from K. Then Kn = conv {x, , . . . , *„} is a random polytope that approximates K (as n -») with high probability. Answering a question of Rolf Schneider we determine, up to first order precision, the expectation of vol K -vol Kn when K is a smooth convex body. Moreover, this result is extended to qu...

Consider a convex domain B of R 3. We prove that there exist complete minimal surfaces which are properly immersed in B. We also demonstrate that if D and D ′ are convex domains with D bounded and the closure of D contained in D ′ then any minimal disk whose boundary lies in the boundary of D, can be approximated in any compact subdomain of D by a complete minimal disk which is proper in D ′. W...