Approximating a compact $C^1$ hypersurface without boundary Can we approximate (arbitrarily closely) a compact $C^1$ hypersurface in Euclidean space without boundary with a polygonal hypersurface, such as a simplicial complex? To clarify, I want to have the $\mathcal{H}^{n-1}$ area of the approximation converge to that of the hypersurface as the diameter of the polygons goes to 0. 
If so, can you please provide a reference for the theorem. Thank you kindly.
Note, I know this to be true for $C^2$ since we have positive reach and can create a tubular neighborhood with the normals. 
Edited: I think I made a mistake to say "same size" above (although that would be ideal for my situation) so I removed it. So in the case of allowing different sizes, can it be done? In fact, I've looked more carefully and they wouldn't have to be the same shape either. The critical piece is to have a piecewise linear surface composed of  $(n−1)$-dimensional polyhedra, meaning the surface is continuous with no "breaks" or "jumps" at the edges, since I will integrate over each piece. Also, there should be a finite number of pieces since the surface is compact, correct? Thanks! 
 A: It seems that what you are asking is essentially how one can approximate the area of a surface via triangulations. This is a classical topic stretching back more than a century. The famous example of Schwartz Lantern shows that arbitrary triangulations do not work. Young first
described conditions which triangulations should satisfy in order to approximate area:
W.H. Young, On the Triangulation Method of Defining the Area of a Surface, 1921.
These approximations have been studied extensively since then. For a description of Young's condition and other references see the following paper posted recently on the arXiv:
Kobayashi and Tsuchiya, Approximating surface areas by interpolations on triangulations, 2017.
See also
L. V. Toralballa, A geometric theory of surface area, 1970
for more references. I think that these papers answer your question. Some of these references are concerned only with 2-D surfaces, but Young's original paper seems to consider arbitrary dimensions. Further, these methods require very little regularity. I am pretty sure that the answer to your question is yes for surfaces in $R^3$, and I think that things should work in all other dimensions as well.
