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A Little on Convexity

this document in PDF: Convexity.pdf

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 $ f$ defined on some interval $ I$ (assuming $ f$ is such that these statements make sense):

  1. All the secants of $ f$ are above the graph of $ f$.
  2. For every $ a,b\in I$ and every $ t\in(0,1)$,

    $\displaystyle f(ta+(1-t)b)<tf(a)+(1-t)f(b). $

  3. The tangents to the graph of $ f$ all lie below that graph and touch it just at the points of tangency.
  4. The derivative $ f'$ is increasing.
  5. The second derivative $ f''$ is positive on $ I$: $ \forall x\in I f''(x)>0$. (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 $ f(x)=x^4$. If all sharp inequalities in this handout are replaced by non-sharp ones (i.e., replace $ >$ by $ \geq$ and $ <$ by $ \leq$ everywhere, with similar corrections for verbal statements), then this statement becomes equivalent to all others).

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

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Dror Bar-Natan 2005-01-31