Concave Growth Function

This is for my homies taking calculus.

Here’s a real-life example of a concave function. Around minute 35, they say that as an animal grows in size, each cell requires less energy — so its total energy consumption is concave as a function of size.

$\dpi{200} \bg_white f( \lambda \cdot x ) \leq \lambda \cdot f(x), \qquad \lambda \in \mathbb{R}^+$

Yup, this is the complicated way to say it — on purpose. Listen to the podcast to hear it in terms of elephants and mice.

I think there is an even more complicated way to put it — maybe it’s called the “Euler characteristic” or something, essentially it’s that

$\dpi{200} \bg_white f(\lambda \cdot x) \approx \lambda^n f(x), \qquad n \in \mathbb{R}$

so that different functions with a different value for n scale in different ways.

If you can put the symbols together with the real-life meaning, it will pay off in terms of insight. You can see something and think to yourself “hey, that’s convex!” And you can also imagine the phenomenon with a different value of n — i.e. different scaling properties.