Geodesic
Lower Bounds for the Square-to-Linear Ratio for Plane Peano Curves
Matematičeskie zametki, Tome 110 (2021) no. 2, pp. 289-296.

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It is proved that, for any map of the unit interval onto the unit square, there exist two points in the interval such that the squared Euclidean distance between their images exceeds the distance between them on the interval at least by a factor of $3.625$. The additional condition that the images of the interval endpoints belong to opposite sides of the square increases this factor to more than $4$.
Keywords: Peano curves, square-to-linear ratio. 
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E. V. Shchepin; E. Yu. Mychka. Lower Bounds for the Square-to-Linear Ratio for Plane Peano Curves. Matematičeskie zametki, Tome 110 (2021) no. 2, pp. 289-296. http://geodesic.mathdoc.fr/item/MZM_2021_110_2_a10/

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