*Published Paper*

**Inserted:** 18 sep 2001

**Last Updated:** 3 feb 2004

**Journal:** Proc. Royal Soc. Edinburgh.

**Volume:** 133

**Pages:** 85-96

**Year:** 2003

**Abstract:**

We provide a simplified proof of the following well-known theorem: {\it The projection to $R^{n+1}$ of a $n$-dimensional rectifiable Legendrian subset of $R^{n+1}\times S^n$ is $C^2$-rectifiable.} Such a result generalizes the following simple fact: {\it Let $M$ be a hypersurface whose Gauss map has graph of class $C^1$. Then $M$ is of class $C^2$.}

**Keywords:**
Geometric measure theory, Legendrian sets