The aim of this paper is to provide an analysis of non-linear approximation in the
$L_p$-norm $p = d / (d - 1)$
of functions of bounded variation on $\mathbb{R}^d$
with $d > 1$
by polynomials in the Haar system. The exponent $p$
is the natural exponent as it is the correct exponent in the Sobolev inequality. The approximation schemes that we
discuss in this paper are mostly related to Haar thresholding and $m$-term
approximation. These problems for $d = 2$ are studied in detail in a paper by Cohen, DeVore, Petrushev and Xu.
The main aim of this paper is to extend their results to the case
$d \geq 2$.
We obtain the optimal order of the $m$-term
Haar approximation and prove the stability of Haar thresholding in the BV-norm. As one of the main tools,
we establish the boundedness of certain averaging projections in BV.