\forall i \in \{0,1\},\\
\rho(\varphi(D_i), \varphi(E_i)) \geq \varepsilon  \ \ \forall i \in \{0,1\}\\
\end{array}\
 \right\}.$$

Actually, this question is a natural generalization of the well-known conjecture that in any compact projective plane, the stabilizer of four points, none three of which are collinear, is compact (see [@Salzmann]\*[Section 44]{}). It should also be emphasized that this question is strongly related to the conjecture mentioned at the beginning of this introduction. Indeed, in order to prove the conjecture, it only remains to show that Condition (ii) or (iii) in Theorem \[theorem:Characterization\] is satisfied by any strongly transitive compact polygon. A positive answer to the above question would ensure that a slightly weaker condition than (iii) holds for all compact polygons; in view of Theorem \[theorem:Characterization\], this would be an important step toward the proof of the conjecture.

The proof of Theorem \[theorem:Criterion\] is given in Section \[section:criterion\], while Theorem \[theorem:Characterization\] is proved in Section \[section:characterization\] (see Corollary \[corollary:MoufangCG\], Proposition \[proposition:proj-CG\] and Theorem \[theorem:Moufang\]). Some of the preparatory results of Section \[section:criterion\] are also used in Section \[section:characterization\] to prove the implication (iii) $\Rightarrow$ (i) of Theorem \[theorem:Characterization\].

Acknowledgement {#acknowledgement .unnumbered}
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The results of this paper are the outcomes of my 