Is it true, that a path connecting two opposite points (i.e. such that the segment joining them passes through the centre of mass of the cube) on the surface of the $d$-dimensional unit cube (with $d>1$) is not shorter than $2$?



Shortest path connecting two opposite points on a cube
Consider the sphere with equator 4.
Divide it into *spherical cubes*, the central projections from an inscribed cube.


Note that the exponential map from tangent plane to the sphere is short.
Note also that if one maps a unit cube centered at the origin by the exponential map it will cover the *spherical cube*.
It is easy to modify the map to get a short symmetric map from the unit cube to each spherical cube.


Joining all these maps, we get a short map fro the surface of the cube to our sphere, where the statement is evident. 


