In case anyone is confused about the terminology, here's where I'm pulling it from:
http://blog.christianperone.com/2011/10/machine-learning-text-feature-extraction-tf-idf-part-ii/
To summarize the relevant part: an L$n$ norm is a notion of distance given by raising each distance component to the $n$th power, summing them, and then taking the $n$th root.
For L1, this simply means to add the components, and it is known as taxicab distance, with physical significance when navigating a grid. For L2, this is the Euclidean distance formula we learned in high school, with obvious physical significance.
What about higher-level norms, though? Is there any physical interpretation of L3, L4, etc., let alone a direct, physical application?