Why does L_1-norm induces sparsity ?

We had a short discussion with Julien Mairal a long while ago and his presentation on Optimization for Sparse Estimation and Structured Sparsity featured at the IMA Short Course entitled ``Applied Statistics and Machine Learning'', reminded me of that conversation on how to easily make sense of the l_1 norm. The video of his talk are here: part I, and part II. Here are the a-ah slides of interest that tells you that only a linear functional will steadily get you to 0.:

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