Note: This question is very similar but different to a previous question [1] of mine. I decided to ask a new question for the sake of clarity.
In order to complete a proof, I need to show the following (explained using figure attached). However, as this seems very intuitive to us, we seek a reference for the below statement (we already have a proof):
Given: We are 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). Additionally, 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.e. the area between the green and the red curve is strictly positive.
Task: Prove that the length of the green line is strictly shorter than the red line. By looking at the picture this is intuitive. This can of course be proven by a lengthy algebraic calculation. However, we want to avoid that and want to refer to a reference for something that intuitive instead.
Question: Can someone think about a reference (not a proof) of this statement? Some notes:
- We need to show that the green curve is strictly shorter than (not just shorter or equal to) the red line
- Based on our literature research, we find references [2] that only prove the "shorter or equal"-version, but not the "strictly shorter"-version.
- As found in [1], this question is related to the Crofton formula. The wikipedia article [3] states that it can be used to show that Between two nested, convex, closed curves, the inner one is shorter. However, no further citation to that statement is given and we can't find one either. Furthermore, I am not sure whether "shorter" means indeed "strictly shorter" or just "shorter or equal to".
Many thanks
[1] Result needed: outside curve longer than convex inside curve
[3] Application section in: https://en.wikipedia.org/wiki/Crofton_formula
The Crofton formula states that the length of a convex curve is proportional to its width. Here, by "width" I mean the average size of its shadow in a random direction. To get a strict inequality, you need to show that there exists some direction along which the green shadow is strictly smaller than the red one. Can you tackle that part of the problem?