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...

The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon’s Conjecture: A hyperbolic group G (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic ...

There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We relate the action of the kernel on the curve complex to a family of actions on trees. Together with a geometric compactness argument we further prove that finitely generated purely pseudo-Anosov subgroups of the kernel are convex cocompact in the sense of Farb and Mosh...

Can you have a non-convex polyhedron with a bigger volume than a convex polyhedron with the isometric surface? Imagine you start blowing air into a polyhedron with a bendable but non-stretchable surface. What can be said about the limiting shape? These are the key questions I will consider. I will start the talk by discussing the history of the problem, presenting examples, and surveying earlie...

In contemporary convex geometry, the rapidly developing Lp-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the Lp-affine surface area for convex bodies. Here, we introduce a functional form of this concept, for log concave and s-concave functions. We show that the new functional form is a generalization of the original Lp-affi...

The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon’s Conjecture: A hyperbolic group G (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic ...

In this paper we prove a maximum principle at infinity for properly embedded surfaces with constant mean curvature H > 0 in the 3-dimensional Euclidean space. We show that no one of these surfaces can lie in the mean convex side of another properly embedded H surface. We also prove that, under natural assumptions, if the surface lies in the slab |x3| < 1/2H and is symmetric with respect to the ...

We suppose that S is a smooth hypersurface in Rn+1 with Gaussian curvature re and surface measure dS, it) is a compactly supported cut-off function, and we let pa be the surface measure with dßa = u>Ka dS. In this paper we consider the case where S is the graph of a suitably convex function, homogeneous of degree d, and estimate the Fourier transform ßa. We also show that if S is convex, with n...

In this paper we consider the problem of efficiently constructing geodesic t-spanners. We consider finding spanners on the surface of a 3 dimensional polyhedron. If Steiner vertices are allowed on the surface of the polyhedron, then we are able to construct sparse t-spanners. If no Steiner vertices are allowed, then we establish lower bounds on the maximum node degree, depending on the spanning...

In the present study, the lenses and ciliary bodies of 20 ostrich eyes were studied macroscopically and microscopically. The histological slides were studied after staining by hematoxylin and eosin, Verhoeff, Van Gieson, and periodic acid-Schiff (PAS). Posterior surface of lens was more convex than its anterior surface. The average lens diameter and thickness were respectively measured as 1.43 ...