# A Little on Convexity

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We are skipping the appendix on convexity of Spivak's Chapter 11, but it is still worthwhile to take something from it (without proof):

Theorem. The following are equivalent, for a function defined on some interval (assuming is such that these statements make sense):

1. All the secants of are above the graph of .
2. For every and every ,

3. The tangents to the graph of all lie below that graph and touch it just at the points of tangency.
4. The derivative is increasing.
5. The second derivative is positive on : . (Gary Baumgartner makes the following correction: This last statement implies all others, but it isn't implied by the others as can be seen by looking for example at . If all sharp inequalities in this handout are replaced by non-sharp ones (i.e., replace by and by everywhere, with similar corrections for verbal statements), then this statement becomes equivalent to all others).

If any of these statements holds, we say that  is convex''. There is a similar theorem with all inequalities reversed, and then the name is  is concave''.

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Dror Bar-Natan 2005-01-31