1. With the loop selected, press <kbd>Alt</kbd><kbd>S</kbd> to scale it
    along the normals.
 2. Press <kbd>S</kbd> again to activate `Even Thickness`, so you define the actual width of the border instead of the corner edges' length.
 3. Enter `-.01` to get a border of 1cm width.

 

The edges don't provide normals to scale along, so the adjacent faces' normals are used, which are pointing inwards. This is why you have to enter a negative distance.