- ...area.
- The
tangent
plane to the reciprocal of
is equivalent to a more awkward tangent
sphere
construction to itself due to Herring[30] that was the first
stability test for surfaces.
- ...true,
- For low symmetry
crystals the Wulff shape is unique, even
though can not be uniquely determined.[42][17]
- ...parts;
- The reason convexification need not apply to elastically
coherent systems is apparent when one considers that a two phase coherent
system can have an free energy that is the sum of the free energies of the
separated phases plus the elastic energy to make them
coherent.[50]
- ...floor.'
-
Note that there are additional areas associated with the two triangles
at the front and back of the tent, but these areas can be made negligible
by making the rectangle very long compared to its width or, equivalently
by corrugating the roof, keeping the orientations fixed, to form
a series of similar small tents, like a `factory roof'.
- ...level.
- In the above
argument we have ignored the energies contributed by edges and corners
separating pieces of planar surface, just as we have ignored the energies of
surfaces between coexisting phases in minimizing
by convexification.
But in the surface case, if such other energies exist they can not be ignored
in forming the limiting factory roof.
- ...conserved.
- The free energy of a system with one mole
should not be compared to the sum of that of two others, each with one
mole.
- ....
- One can create a meaningful inequality for by lifting
this HD1 function from the chord to the surface of the unit sphere, but
a simpler method is developed here.
- ...Discussion
is the simplex:
where
, in two-dimensions
it is a line-segment; in three dimensions an equilateral
triangle, etc.
The chemical Wulff construction
could as well have been written as
,
where 
the portion of the unit sphere embedded
in 