Result needed: outside curve longer than convex inside curve

Question

In order to complete a proof, I need to show the following (explained using figure attached):

I am given two points A and B in the two-dimensional plane connected by the direct black line. Points A and B are also connected by the green vectors. The corresponding green curve is convex. Finally, points A and B are also connected by the red vectors.

The corresponding red curve

• is convex
• goes from A to B "on the same side" as the green curve (with respect to the black line)
• goes "outside" the green curve (with respect to the black line)

I want to prove that the length of the red line is greater or equal to the green line. By looking at the picture this is obvious. In order to formally prove this, I could use a lengthy algebraic calculation. However I feel that what I want to prove should follow immediately from some result from the field of analysis for example (of which I am not an expert).

Can someone think about a theorem that immediately implies that the red curve is at least as long as the green curve?

Many thanks!

Answer 1

Apparently this follows from the Crofton formula : Take a look at the "applications" section of the Wikipedia page: Crofton formula

I found this from this MO question and answer.

Answer 2

By several straight cuts you can get from the red piece to the inside green piece. At each step the perimeter of the piece is decreased, since a broken line is replaced with a straight segment.