TITLE: Can a convex polygon inside a square with edge length 1 have a perimeter > 4?
QUESTION [6 upvotes]: While featherbrainedly doodling the other day I noticed that it's probably impossible to draw a convex polygon with a greater perimeter then that of the square around it.
Can someone find a counterexample or maybe even find a proof?

REPLY [6 votes]: I'll rephrase the argument by Emanuele Paolini without "$1$-Lipschitz" and projection. Place a bomb inside of the convex polygon. When it goes off, the polygon breaks apart at the vertices, with every side of it flying out in the direction in which it was facing: I drew   arrows to show the directions in which two of the sides go.

Flying out, each side  takes out a chunk of the boundary of the square. Use some geometry to prove that

The chunk taken out by each side is at least as long as the side itself. 
The chunks taken out by different sides do not overlap.

Conclusion follows from 1 and 2.