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We are given a disk $D$ in $\mathbb{R}^2$. Let $C$ be its boundary (i.e., the circle bounding $D$ on its plane). Let $P(n,d)$ be a set of $n$ pairs of chords of $C$ such that for each $\{c,c'\}\in P(n,d)$, $c$ and $c'$ are parallel and the distance between them is equal to $0<d<r$, where $r$ is the length of the radius of $C$.

Finally, let $R$ be the set of all connected regions of $D$ obtained by cutting $D$ by all pairs of chords of $P(n,d)$, in such a way that each region of $A\in R$

  • does not contain any point belonging to any of such chords,
  • is bounded by some of these chords or $C$, and
  • there exists a pair of chords $\{c,c'\}\in P(n,d)$ such that all points of $A$ are located between $c$ and $c'$.

Question: What is the maximum number $m(n,d)$ of regions in $R$ over all possible set of $P(n,d)$?

Case n=3 Example: For all $d$, we have $m(1,d)=1$, $m(2,d)=5$, $m(3,d)=13$ - In fact, I actually conjecture that $m$ depends on $n$ solely.

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    $\begingroup$ This may be just $A058331-2n$ (oeis.org/A058331) or $2n^2-2n+1$, if cutting regions by parallel lines is the same as cutting regions by hyperbolas, and there are always $2n$ regions to remove as being outside all the lines. $\endgroup$
    – user44143
    Commented Mar 11, 2021 at 13:35
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    $\begingroup$ The maximum number of cells in an arrangement of $n$ lines is $n(n+1)/2+1$. So your $2n$ lines could make at most $2n^2+n+1$ cells, if all were in $D$. So indeed there is a quadratic upper bound. MattF was seeking an exact formula. $\endgroup$
    – Joseph O'Rourke
    Commented Mar 11, 2021 at 14:05
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    $\begingroup$ @MattF.: Nice conjecture! Could you explain the intuition behind: hyperbolas = parallel lines? $\endgroup$
    – Joseph O'Rourke
    Commented Mar 11, 2021 at 14:15
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    $\begingroup$ Well, topologically the hyperbolas are the same as pairs of parallel lines, and the configuration of the plane most cut up by two hyperbolas is topologically the same as the configuration most cut up by two pairs of parallel lines. $\endgroup$
    – user44143
    Commented Mar 11, 2021 at 14:32
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    $\begingroup$ Thank you, I see I misunderstood the last condition -- c and c' need not to meet $\partial A$ $\endgroup$
    – Pietro Majer
    Commented Mar 11, 2021 at 18:44

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