I'm learning about the Statistical learning and in the section comparing Lasso and Ridge Regression it shows that the main difference between these two problems is the way the constraint/penalty is formulated.
In Lasso, the penalty is $\ell_1$ norm: $\lambda \sum |\beta_j|$, while in regression, the penalty is $\ell_2$: $\lambda \sum \beta_j^2$.
Geometrically, this means that the lasso will have a constraint in the form of a diamond (in 2 dimensions), and in higher dimensions it will have vertices and edges. For ridge regression, in 2D, it is a circle, and hypersphere in higher dimensions.
My question is: The author claims that you get SPARSITY in the lasso. I do not understand why, even with the geometric picture above. And what is the clear advantage of Lasso over ridge regression?
Your insights would be very valuable. I appreciate if your answer would contain some mathematics, but more importantly, intuition. Thanks