  We focus on the high dimensional linear regression
$Y\sim\mathcal{N}(X\beta^{*},\sigma^{2}I_{n})$, where
$\beta^{*}\in\mathds{R}^{p}$ is the parameter of interest. In this setting,
several estimators such as the LASSO and the Dantzig Selector are known to
satisfy interesting properties whenever the vector $\beta^{*}$ is sparse.
Interestingly both of the LASSO and the Dantzig Selector can be seen as
orthogonal projections of 0 into
$\mathcal{DC}(s)=\{\beta\in\mathds{R}^{p},\|X'(Y-X\beta)\|_{\infty}\leq s\}$ -
using an $\ell_{1}$ distance for the Dantzig Selector and $\ell_{2}$ for the
LASSO. For a well chosen $s>0$, this set is actually a confidence region for
$\beta^{*}$. In this paper, we investigate the properties of estimators defined
as projections on $\mathcal{DC}(s)$ using general distances. We prove that the
obtained estimators satisfy oracle properties close to the one of the LASSO and
Dantzig Selector. On top of that, it turns out that these estimators can be
tuned to exploit a different sparsity or/and slightly different estimation
objectives.
