  It is proved that for any mapping of a unit segment to a unit square, there
is a pair of points of the segment for which the square of the Euclidean
distance between their images exceeds the distance between them on the segment
by at least $3\frac58$ times. And the additional condition that the images of
the beginning and end of the segment belong to opposite sides of the square
increases the estimate to $4+\varepsilon$.
