Talk:Power Function is Convex Real Function

Probably need a better name for this, seeing as it clashes with Power Function on Strictly Positive Base is Convex. Caliburn (talk) 18:38, 6 January 2023 (UTC)

They're effectively the same thing, so they should be merged. Anyone up for it? --prime mover (talk) 20:04, 6 January 2023 (UTC)

They're a bit different, the first is basically about $x \mapsto e^x$, this is about $x \mapsto x^2$, $x \mapsto x^\pi$ and similar. You could compose with a log to get this result, though. Caliburn (talk) 20:09, 6 January 2023 (UTC)

They are all just examples of the same truth: $a \to b^c$ is convex in general. Within certain constraints, of course, which are to be presented in the conventional way. --prime mover (talk) 20:29, 6 January 2023 (UTC)

I don't get the importance of this particular result. Why is the strict convexity of $x^p$ in case of $p>1$ is not mentioned? Why not more generally state Real Function with Positive Second Derivative is Convex? --Usagiop (talk) 22:30, 6 January 2023 (UTC)

Because it hasn't been posted. Well volunteered, that man. --prime mover (talk) 23:10, 6 January 2023 (UTC)

Throwing this up as a lemma for the result that if $\tuple {X, \Sigma, \mu}$ is a finite measure space and $p \ge q \ge 1$ then $\map {\LL^p} {X, \Sigma, \mu} \subseteq \map {\LL^q} {X, \Sigma, \mu}$. Will put this up tomorrow. Also in my PDEs class I always found myself appealing to the fact that $x \mapsto \size x^p$ is convex. Caliburn (talk) 22:57, 6 January 2023 (UTC)