On geometric characteristics of convex functions and Laplace integrals

In many problems of analysis the second derivatives are used to characterize the convexity of the function that imposes serious restrictions on a class of considered functions. This paper introduces the geometric characteristics of convexity, which from our point of view are more natural in the study of weighted spaces of functions. In the one-dimensional case, the problem is considered in more detail and we define the various characteristics, which are in a sense equivalent. As an application we study the asymptotic behavior of multidimensional Laplace integral.