Geodesic
On Fractal Peano Curves
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 294-303

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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). 
@article{TM_2004_247_a22,
     author = {E. V. Shchepin},
     title = {On {Fractal} {Peano} {Curves}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {294--303},
     publisher = {mathdoc},
     volume = {247},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_247_a22/}
}
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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/