Two related problems concerning continuous functions on a sphere Sn−1 ⊂ Rn are studied, together with the problem of finding a family of polyhedra in Rn one of which is inscribed in (respectively, circumscribed about) a given smooth convex body in Rn. In particular, it is proved that, in every convex body K ⊂ R3, one can inscribe an eight-vertex polyhedron obtained by “equiaugmentation” of a si...

We estimate the number and ratio of negative homothetic copies of a d-dimensional convex body C sufficient for the covering of C. If the number of those copies is not very large, then our estimates are better than recent estimates of Rogers and Zong. Particular attention is paid to the 2-dimensional case. It is proved that every planar convex body can be covered by two copies of ratio −4 3 (thi...

We provide an affirmative answer to a problem posed by Barvinok and Veomett in [4], showing that in general an n-dimensional convex body cannot be approximated by a projection of a section of a simplex of sub-exponential dimension. Moreover, we prove that for all 1 ≤ n ≤ N there exists an n-dimensional convex body B such that for every n-dimensional convex body K obtained as a projection of a s...

Many crucial results of the asymptotic theory of symmetric convex bodies were extended to the non-symmetric case in recent years. That led to the conjecture that for every n-dimensional convex body K there exists a projection P of rank k, proportional to n, such that PK is almost symmetric. We prove that the conjecture does not hold. More precisely, we construct an n-dimensional convex body K s...

Let B be an o-symmetric convex body in R, and M be the normed space with unit ball B. The M-thickness ∆B(K) of a convex body K ⊆ R is the smallest possible Mdistance between two distinct parallel supporting hyperplanes of K. Furthermore, K is said to be M-reduced if ∆B(K ′) < ∆B(K) for every convex body K ′ with K ′ ⊆ K and K ′ 6= K. In our main theorems we describe M-reduced polytopes as polyt...

We present numerical methods to solve optimization problems on the space of convex functions or among convex bodies. Hence convexity is a constraint on the admissible objects, whereas the functionals are not required to be convex. To deal with, our method mix geometrical and numerical algorithms. We give several applications arising from classical problems in geometry and analysis: Alexandrov’s...

We prove an embedding theorem that says a convex body can pass through a triangular hole∆ if and only if the convex body can be congruently embedded in a right triangular prism with base ∆. Combining this with a known result on congruent embeddings of a regular tetrahedron in a triangular prism, we show that a regular tetrahedron with unit edge can pass through an equilateral triangular hole in...

Objective: To research on the application of high dimensional space theory to the biomimetic pattern recognition. Procedures: The sample was constructed into a k-dimensional simplex in a high dimensional space and 0.85 times of the average distance between the vertices was chosen as the threshold value thus a convex cell body covering the k-dimensional simplex was constructed. By determining wh...

We propose a general method to determine the theoretical microstructure in one dimensional elastic bars whose internal deformation energy is given by non-convex polynomials. We use non-convex variational principles and Young measure theory to describe the optimal energetic configuration of the body. By using convex analysis and classical characterizations of algebraic moments, we can formulate ...

A random polytope is the convex hull of uniformly distributed random points in a convex body K. A general lower bound on the variance of the volume and f -vector of random polytopes is proved. Also an upper bound in the case when K is a polytope is given. For polytopes, as for smooth convex bodies, the upper and lower bounds are of the same order of magnitude. The results imply a law of large n...