We have two convex sets $S,T$ such that $S\subseteq T$. Prove or disprove that the circumference of $S$ must be smaller than that of $T$.

Asked May 17 '22 at 03:49

Active May 17 '22 at 03:49

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This question originated from an exercise asking to compare the arclength of $y=x^2$ between $(0,0)$ and $(1,1)$ and $\frac{\pi}{2}$.

The solution starts by constructing the circle with center $(0,1)$ and radius $1$, and showing that the arc of parabola sits entirely within the circle, thus the arclength is less than $\frac{\pi}{2}$.

However, I'm not entirely sure that this is a valid proof, for the last step invokes some geometric intuition and I do not see an obvious way to rephrase it rigorously.

asked May 17 '22 at 03:49

pzq_alex

Proof by Buffon's noodle. Since $S\subseteq T$, the expected number of intersections with grid lines must have $\partial S\le\partial T$, so the lengths also follow $\partial S\le\partial T$. – Edward H May 17 '22 at 04:02
How do you define the circumference? – copper.hat May 17 '22 at 06:39

We have two convex sets $S,T$ such that $S\subseteq T$. Prove or disprove that the circumference of $S$ must be smaller than that of $T$.

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