Geodesic
On Fractal Peano Curves
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 294-303 
E. V. Shchepin. On Fractal Peano Curves. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 294-303. http://geodesic.mathdoc.fr/item/TM_2004_247_a22/
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     title = {On {Fractal} {Peano} {Curves}},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

It is shown that, for a fractal Peano curve $p(t)$ that maps a unit segment onto a unit square, there always exists a pair of points $t,t'$ of the segment that satisfy the inequality $|p(t)-p(t')|^2\ge 5|t-t'|$. As is clear from the classical Peano–Hilbert curve, the number $5$ in this inequality cannot be replaced by a number greater than $6$ (the result of K. Bauman).

[1] Schepin E. V., “Povyshayuschie razmernost otobrazheniya i nepreryvnaya peredacha informatsii”, Voprosy chistoi i prikladnoi matematiki, v. 1, Priok. kn. izd-vo, Tula, 1987, 148–155

[2] Bauman K. E., Koeffitsient rastyazheniya krivoi Peano–Gilberta, , 2004 http://www.mi.ras.ru/~scepin/rast6.pdf

[3] Schepin E. V., Bauman K. E., “O krivykh Peano fraktalnogo roda 9”, Modelirovanie i analiz dannykh, Tr. fak. inform. tekhnol. MGPPU., no. 1, RUSAVIA, M., 2004, 79–89