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Let $f: P^4 \to \mathbb R^+$, $(A,B,C,D)\mapsto (AB-BC)^2+(BC-CD)^2+(CD-DA)^2+(AC-BD)^2$. Where $AB$ is the distance betwin $A$ and $B$.

Let $A$, $B$, $C$, $D$ be four points of the plane $P$, respectively in $S_A$, $S_B$, $S_C$, $S_D$ that are areas delimited by squares. We also suppose that no point is the corner of its square—but it can be on the egde.

Suppose that $f((A,B,C,D))>0$ (then $ABCD$ is not a square), does there always exist $A'\in S_A$, $B'\in S_B$, $C'\in S_C$, $D'\in S_D$ such that $f((A',B',C',D'))<f((A,B,C,D))$?

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  • $\begingroup$ I found the quantification a little ambiguous as stated it, so I changed it, I think while preserving your meaning. Please feel free to undo it if I failed (although I encourage you to keep the spell-checking). Also, do $S_A$, $S_B$, $S_C$, $S_D$ have to be disjoint (or at least different)? $\endgroup$
    – LSpice
    Commented Apr 27, 2020 at 8:15
  • $\begingroup$ Thank you very much LSpice, it is much better this way, with the definition of $f$ at the beginning. Maybe there is still ambiguities, indeed, i ask if it is true for any $S_A,S_B,...$ (typically as small as we want) . $\endgroup$
    – jcdornano
    Commented Apr 27, 2020 at 9:15
  • $\begingroup$ What does $AB$ mean, when $A$ and $B$ are points in a plane? $\endgroup$
    – Gerry Myerson
    Commented Apr 27, 2020 at 12:46
  • $\begingroup$ This is a stadard notation in France for the distance betwin $A$ abd $B$ , but must admit i never met it in any graduate text, I'm going to edit $\endgroup$
    – jcdornano
    Commented Apr 27, 2020 at 22:59
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    $\begingroup$ I'm curious about the close votes. This looks to me like a question about elementary subject matter, but it's not clear that it's an elementary question—at least, I don't see how to solve it. If it's trivial, then I think it might be a good idea for a close-voter to say so. $\endgroup$
    – LSpice
    Commented Apr 28, 2020 at 15:47

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