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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
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| ``Re: Why convex?''
by mps on 2007-02-26 10:21:32 |
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| > Why is this called "convexity conjecture"? I've read the
> reference, too, and they don't explain it at all.
Here's a first approximation: if $\pi$ were a convex function,
then it would follow that
\[
\pi( (x+y)/2 ) \le (\pi(x) + \pi(y)) / 2.
\]
But \pi( (x+y)/2 ) doesn't make sense if $x+y$ is odd, so
let's drop the 2s:
\[
\pi(x + y) \le \pi(x) + \pi(y).
\]
I'm a bit uneasy about this, since $\pi(x/2)$ is not the same
as $\pi(x)/2$, but that may also be the reason Crandall and
Pomerance put ``convexity'' in scare quotes.
I hope an analyst will pop in to clear things up. |
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