Functions of bounded variation are not dense in the Strong BD Norm

Abstract

We prove that functions of bounded variation are not dense in the space of functions of bounded deformation with respect to the strong BD norm. Using alternating Riesz products and directional slicing, we construct on every nonempty open subset of ${\mathbb{R}}^n$ ($n\ge 2$) a continuous, compactly supported BD function that is at positive distance from BV.