###### tags: `one-offs` `convexity`

# Log-Convexity of the $C^\alpha$ Semi-Norm

**Overview**: In this note, I give a quick proof of a nice convexity result.

## From Moment-Generating Functions to Uniform Continuity Constants

A well-known fact in the context of exponential family theory (and / or statistical mechanics) is that for suitable functions $F$, the cumulant-generating function

\begin{align}
A:\theta\mapsto\log\left(\int\mu\left(\mathrm{d}x\right)\cdot\exp\left(\left\langle \theta,F\left(x\right)\right\rangle \right)\right)
\end{align}

is convex, modulo a few standard conditions. Less well-known (but perhaps indirectly related?) is that the logarithm of the $C^{\alpha}$ seminorm of a function is also convex! That is, fix a function $F$ and define

\begin{align}
H:\alpha\mapsto\sup\left\{ \frac{\mathsf{d}_{\mathcal{Y}}\left(F\left(x\right),F\left(x^{'}\right)\right)}{\mathsf{d}_{\mathcal{X}}\left(x,x^{'}\right)^{\alpha}}:x,x^{'}\in\mathcal{X},x\neq x^{'}\right\} .
\end{align}

It then holds that

\begin{align}
\log H\left(\alpha\right)=\sup\left\{ \log\mathsf{d}_{\mathcal{Y}}\left(F\left(x\right),F\left(x^{'}\right)\right)-\alpha\cdot\log\mathsf{d}_{\mathcal{X}}\left(x,x^{'}\right):x,x^{'}\in\mathcal{X},x\neq x^{'}\right\} ,
\end{align}

which is a supremum of functions which are affine in $\alpha$, and is hence convex in $\alpha$. I'm not sure about whether this has any serious applications, but it seems neat! It would also be rewarding to see whether there is a genuine connection to the result about cumulant-generating functions.