Convexity is a classical idea. Archimedes used a version of convexity in his considerations of arc length. Yet the idea was not formalized until 1934 in the monograph of Bonneson and Fenchel [BOF]. The classical definition of convexity is this: An open domain Ω ⊆ R is convex if, whenever P,Q ∈ Ω, then the segment PQ connecting P to Q lies in Ω. We call this the synthetic definition of convexity...

Shape descriptors are used in many computer vision tasks. Convexity is one of the most widely used shape descriptors in practical applications and also one of the most studied in the literature. There are already several defined convexity measures. The most standard one comes from the comparison between a given shape and its convex hull, but there also other approaches. Independently of whether...

Recently, Došlić, and Liu and Wang developed techniques for dealing with the log-convexity of sequences. In this paper, we present a criterion for the log-convexity of some combinatorial sequences. In order to prove the log-convexity of a sequence satisfying a three-term recurrence, by our method, it suffices to compute a constant number of terms at the beginning of the sequence. For example, i...

A model for a set of stock prices is said to be convexity preserving if the price of any convex European claim is convex as a function of the underlying stock prices at all times prior to maturity. As is well-known, this property is intimately connected to certain monotonicity properties of the option price with respect to volatility and other parameters of the model. Generally speaking, if the...

The monotonicity, convexity, log-convexity and complete monotonicity properties for some special functions are proved, and some inequalities are established.

In this work we investigate the behavior of various geodesic convexity parameters with respect to the Cartesian product operation for graphs. First, we show that the convex sets arising from geodesic convexity in a Cartesian product of graphs are exactly the same as the convex sets arising from the usual binary operation ⊕ for making a convexity space out of the Cartesian product of any two con...