This post continues the theme of operating with functions using their graphs. Given an integrable function on the interval , consider the region bounded by the graph , the axis , and the vertical lines , .
The area of is exactly , the norm of . On the other hand, the area of a set is the integral of its characteristic function,
So, the correspondence is a map from the space of integrable functions on , denoted , to the space of integrable functions on the plane, denoted . The above shows that this correspondence is norm-preserving. It also preserves the metric, because integration of gives the area of the symmetric difference , which in turn is equal to . In symbols:
The map is nonlinear: for example is not mapped to (the function that is equal to 2 on the same region) but rather to a function that is equal to 1 on a larger region.
So far, this nonlinear embedding did not really offer anything new: from one space we got into another. It is more interesting (and more difficult) to embed things into a Hilbert space such as . But for the functions that take only the values , the norm is exactly the square root of the norm. Therefore,
In other words, raising the metric to power creates a metric space that is isometric to a subset of a Hilbert space. The exponent is sharp: there is no such embedding for the metric with . The reason is that , having the Manhattan metric, contains geodesic squares: 4-cycles where the distances between adjacent vertices are 1 and the diagonal distances are equal to 2. Having such long diagonals is inconsistent with the parallelogram law in Hilbert spaces. Taking the square root reduces the diagonals to , which is the length they would have in a Hilbert space.
This embedding, and much more, can be found in the ICM 2010 talk by Assaf Naor.