# Norms in differential privacy

I know that perturbation should be proportional to the $$\text{L}_1$$-sensitivity of the function if someone wants ($$\epsilon,0$$)-differential privacy, and proportional to the $$\text{L}_2$$-sensitivity of the function if someone wants ($$\epsilon, \delta$$)-differential privacy for $$\delta > 0$$.

But why is that? Can anybody describe the intuition behind the use of these two norms? Or maybe share some resources?

I don't think there is a direct between the type of norm and whether $$\delta=0$$. For example, this crypto.SE answer mentions a mechanism that is pure $$\varepsilon$$-DP and scales with $$L_2$$-sensitivity. Conversely, some $$(\varepsilon,\delta)$$-DP mechanisms scale with $$L_1$$-sensitivity, like the truncated geometric distribution from this paper.
However, the most commonly used noise mechanisms used in DP are Laplace noise and Gaussian noise, which respectively scale with $$L_1$$ and $$L_2$$ sensitivity; the first one is pure $$\varepsilon$$-DP while the latter only provides $$(\varepsilon,\delta)$$-DP. You can find a "visual" representation of why that's the case by looking at the density function of a multi-dimensional Gaussian (top) vs. Laplace (bottom):