We show that a PL-realization of a closed connected manifold of dimension n − 1 in R (n ≥ 3) is the boundary of a convex polyhedron if and only if the interior of each (n− 3)-face has a point, which has a neighborhood lying on the boundary of a convex n-dimensional body. This result is derived from a generalization of Van Heijenoort’s theorem on locally convex manifolds to the spherical case. N...

AIM To evaluate the image quality produced by polypseudophakia used for strongly hypermetropic and nanophthalmic eyes. METHODS Primary aberration theory and ray tracing analysis were used to calculate the optimum lens shapes and power distribution between the two intraocular lenses for two example eyes: one a strongly hypermetropic eye, the other a nanophthalmic eye. Spherical aberration and ...

knowing the fact that the main weakness of the most standard methods including k-means and hierarchical data clustering is their sensitivity to initialization and trapping to local minima, this paper proposes a modification of convex data clustering  in which there is no need to  be peculiar about how to select initial values. due to properly converting the task of optimization to an equivalent...

‎In this paper‎, ‎we study the convex combinations of harmonic mappings obtained by shearing a class of slit conformal mappings‎. ‎Sufficient conditions for the convex combinations of harmonic mappings of this family to be univalent and convex in the horizontal direction are derived‎. ‎Several examples of univalent harmonic mappings constructed by using these methods are presented to illustrate...

In this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. Also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. Furthermore,...

A Gaussian polytope is the convex hull of normally distributed random points in a Euclidean space. We give an improved error bound for the expected number of facets of a Gaussian polytope when the dimension of the space is fixed and the number of points tends to infinity. The proof applies the theory of the asymptotic distribution of the top order statistic of a collection of independently dist...

Convex hulls are fundamental geometric tools used in a number of algorithms. This paper presents a fast, simple to implement and robust Smart Convex Hull (S-CH) algorithm for computing the convex hull of a set of points in . This algorithm is based on “spherical” space subdivision. The main idea of the S-CH algorithm is to eliminate as many input points as possible before the convex hull constr...

Let p be a positive number. Consider the probability measure γ p with density ϕ p (y) = c n,p e − |y| p p. We show that the maximal surface area of a convex body in R n with respect to γ p is asymptotically equivalent to C (p)n 3 4 − 1 p , where the constant C (p) depends on p only. This is a generalization of results due to Ball (1993) [1] and Nazarov (2003) [9] in the case of the standard Gau...

In this article we discuss lattice convexity and concavity of Calderón-Lozanovskii space Eφ , generated by a quasi-Banach space E and an increasing Orlicz function φ. We give estimations of convexity and concavity indices of Eφ in terms of Matuszewska-Orlicz indices of φ as well as convexity and concavity indices of E. In the case when Eφ is a rearrangement invariant space we also provide some ...