Strict convexity of Ridge vs Convexity of LASSO Is there any intuition why the ridge regression is strictly convex, while the LASSO is only convex? 
Does it have to do with the "corners" of the L1 regularization? 
 A: Conceptually, a function is convex is for any pair $(x_1, x_2)$, the line segment joining $(x_1,f(x_1))$ and $(x_2,f(x_2))$ lies above the curve $y=f(x)$. It is strictly convex if this line segment strictly lies above the curve (i.e. the only points they have in common are the endpoints $(x_1,f(x_1))$ and $(x_2,f(x_2))$). The LASSO penalty is not strict, because if $x_1$ and $x_2$ have the same sign, then the line segment and the curve are exactly equal, and therefore they have infintely many points in common.
A: You need to begin with an understanding of what a "strictly convex" function is.  A function f(x) is strictly convex on a convex domain if for every $x_{1}$ and $x_{2}$ in the domain and every $t$, with $0< t < 1$, 
$ f((1-t)x_{1}+tx_{2}) < (t-1)f(x_{1})+tf(x_{2})$
In LASSO, you're minimizing $\| \beta \|_{1}$, which is not a strictly convex function.  
In ridge regression, you're minimizing $\| X \beta -y \|_{2}^{2}+\lambda \| \beta \|_{2}^{2}$, which is a strictly convex function.  
