While thinking about Lorenz curves and economic inequality I wondered if you can extend Lorenz curves to Lorenz surfaces by revolving a Lorenz curve about the line of perfect equality.
Would such a Lorenz surface have any economical interpretation?
Can Lorenz curves be analytic functions, such as $y=x^2?$
Here is a picture of the Lorenz surface I'm imagining inside a cube.
The line of perfect equality would now be the longest diagonal of the cube.