An important class of planar straight-line drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decomposition of a biconnected graph into its triconnected components. We then consider the problem of te...

The paper is a supplement to [2]. Let L be a lattice and U an o-symmetric convex body inR n. The Minkowski functional I[ I1~ of U, the polar body U ~ the dual lattice L*, the covering radius/z(L, U), and the successive minima ~.i(L, U), i = 1 . . . . . n, are defined in the usual way. Let 12, be the family of all lattices in R". Given a convex body U, we define mh(U) = sup max ~.i(L, U)~.n-i+l(...

We study the conditions under which one is able to efficiently apply variancereduction and acceleration schemes on finite sum optimization problems. First, we show that, perhaps surprisingly, the finite sum structure by itself, is not sufficient for obtaining a complexity bound of Õ((n + L/μ) ln(1/ )) for L-smooth and μ-strongly convex individual functions one must also know which individual fu...

We study the conditions under which one is able to efficiently apply variance-reduction and acceleration schemes on finite sum optimization problems. First, we show that, perhaps surprisingly, the finite sum structure by itself, is not sufficient for obtaining a complexity bound of Õ((n + L/μ) ln(1/ǫ)) for L-smooth and μ-strongly convex individual functions one must also know which individual f...

We demonstrate a new platform, convex lens-induced nanoscale templating (CLINT), for dynamic manipulation and trapping of single DNA molecules. In the CLINT technique, the curved surface of a convex lens is used to deform a flexible coverslip above a substrate containing embedded nanotopography, creating a nanoscale gap that can be adjusted during an experiment to confine molecules within the e...

Abstract We develop a structure theory for $$\mathrm {RCD}$$ RCD spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is topological manifold boundary whose interior equal to the set of regular points. Further points smooth and geodesically convex. Around there a...

A function f is said to be cone superadditive if there exists a partition of R into a family of polyhedral convex cones such that f(z + x) + f(z + y) ≤ f(z) + f(z + x+ y) holds whenever x and y belong to the same cone in the family. This concept is useful in nonlinear integer programming in that, if the objective function is cone superadditive, the global minimality can be characterized by loca...

New results on approximation of a convex body K ⊂ R3 by affine images of circular cylinders, parallelepipeds, hexagonal and octagonal regular (and some other) prisms are obtained. Two of the theorems obtained are as follows (V (K) denotes the volume of a body K ⊂ R3). Theorem 1. Let K be an arbitrary convex body in R3. There exists a regular octagonal prism an affine image of which is circumscr...

This paper is to characterize L-concavities and L-convexities via some derived forms of relations operators. Specifically, notions L-concave internal relation space hull are introduced. It proved that the category L-concave
 spaces isomorphic spaces. Also, L-convex enclosed