Intuitive explantion: What is a Finsler metric?

Neither of the two most evident sources,

seems to provide me with the high-level intuition that I could convey to students in ~10 minutes. And that is because I myself am not sufficently schooled in Finsler geometry.

Either a pointer to a better source, our your own take, would be welcomed. Thanks!

• Perhaps you would enjoy ams.org/notices/199609/chern.pdf – Steven Gubkin Sep 11 '14 at 1:15
• @StevenGubkin: Perfect, Steven---Thanks! – Joseph O'Rourke Sep 11 '14 at 1:18
• So, is the answer simply the title of the linked article. I think that comes in at about 10 seconds, well short of the 10 minute mark. – James S. Cook Sep 11 '14 at 5:00
• Why do you want to teach either students or yourself about Finsler metrics? – user173 Sep 12 '14 at 4:33

$$\int \sqrt[\large{4}]{\dot{x}^4 + \dot{y}^4} dt.$$