# PDF Convexity and Graph Theory

Hiroshi Hirai Author information. Hiroshi Hirai The University of Tokyo.

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Article overview. References Related articles 0. Figures 0. Information related to the author. Written by top experts, these papers highlight the close connections between these fields, as well as ties to other domains of geometry and their reciprocal influence. They offer an overview on recent developments in geometry and its border with discrete mathematics, and provide answers to several open questions.

The volume addresses a large audience in mathematics, including researchers and graduate students interested in geometry and geometrical problems. Toggle navigation. In probability theory , a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable.

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The concept of strong convexity extends and parametrizes the notion of strict convexity. A strongly convex function is also strictly convex, but not vice versa. Some authors, such as  refer to functions satisfying this inequality as elliptic functions.

## Convexity and Graph Theory, Proceedings of the Conference on Convexity and Graph Theory

An equivalent condition is the following: . It is not necessary for a function to be differentiable in order to be strongly convex.

Using Taylor's Theorem there exists. The distinction between convex, strictly convex, and strongly convex can be subtle at first glance.

The proof of this statement follows from the extreme value theorem , which states that a continuous function on a compact set has a maximum and minimum. Strongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class.

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