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Consider a planar convex region $C$ of unit area and all chords of it that cut off a segment of area $\alpha$ from $C$. Obviously, if $C$ is a circular disk of unit area, all segments of area $\alpha$ cut from it by chords are congruent for any value of $\alpha$.

Question: Is there any unit area $C$ other than the circular disk with the property: for even any one value of the fraction $\alpha$, all segments of area $\alpha$ cut from $C$ by its chords will also have equal perimeter?

Note 1: The answer seems to be "No" but I have no proof. And if the answer is "yes", one could ask if some stronger conditions on the equal area segments will pick the circular disk as the only possible $C$.

Note 2: Question could be naturally generalized to higher dimensions and in 2D, 'perimeter' replaced by say, 'diameter' or even 'moment of inertia about the partitioning chord'. Following comments from Pierre and Ilya Bogdanov below, one could also ask a variant with "equal perimeter" replaced by "equal length of the outer boundary of $C$" wherein $\alpha$ = 1/2 is degenerate.

An argument for "No" as answer to the basic version of question: Every segment cut by a chord from any $C$ consists of two parts - the straight chord and a curved piece of the boundary. From any unit area $C$, Consider a segment $S$ with area $\alpha$ cut by a chord and compare it to a segment of area $\alpha$ cut from a unit circle - call this segment $S_0$. The chord part of $S$ has to be shorter than the chord of $S_0$ (otherwise, the area of $S$ will be less than $S_0$ by Dido's problem) and hence the curved part of S will be more than that of $S_0$. So, roughly speaking, the curve part of $S$ will bulge outwards more from its chord than the circular arc part of $S_0$ bulges from its chord. It seems a bulge greater than that of the circular arc cannot be sustained for the entire boundary of $C$ (for every segment of area $\alpha$ cut from $C$) without $C$ becoming non-convex. I can't make more rigorous this notion of 'local convexity' of a convex region that could be defined in terms of this bulge.

Nandakumar R
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  • @PierrePC No, since the perimeter of the segment includes the chord length. Your example would work only in the case when all central chords have equal lengths. – Ilya Bogdanov Oct 26 '23 at 08:56
  • Thanks. My question was about the full perimeter of the segments, including the chord. But then, a variant can probably be asked with the condition "all chord-separated segments of C with area alpha have equal length of the outer boundary of C". – Nandakumar R Oct 26 '23 at 11:01
  • Pierre, your observation was actually quite relevant. It broadened the question and made me realize that this is actually 3 questions in one - convex regions with the property that all chord-cut segments of area alpha also having (a) equal total perimeter OR (b) equal portion of outer boundary OR (c) equal chord length. They may not all be independent questions but I have no clear idea. Thanks! Indeed, one can bring in other quantities such as diameter and generate further questions! – Nandakumar R Nov 01 '23 at 14:48

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