TITLE: "a shape that ... lies halfway between a square and a circle"
QUESTION [20 upvotes]: An article in the
Notices of the AMS, Volume 61, Issue 10, 2014
(PDF download link),
on Khot's Unique Games Conjecture, says this:

Another group ... found a 
  shape that in a certain sense lies halfway between 
  a square and a circle (though in many more than 
  two dimensions). 
  Like a square, copies can be placed next to 
  each other horizontally and vertically to fill a 
  whole space without gaps or overlaps, forming a 
  multi-dimensional foam. But its perimeter is much 
  smaller than a square—it’s closer to that of a circle, 
  the object which has the smallest perimeter for the 
  area contained.

Could someone please either explain this enigmatic description further,
or provide a reference to the original work? Thanks!
(This issue is related to my earlier [answered] question,
"Optimal planar net for catching convex shapes.")

REPLY [12 votes]: Just to supplement Carlo's answer with a figure
that illustrates this intermediate cube-sphere shape in $\mathbb{R}^3$, 
from the 2012 paper
he cites—"we provide the most 
efficient known cubical foam in 3 dimensions":