The L1 Poincare inequality – bounding the 1-norm of a function by the 1-norm of it’s derivative – can be understood as a consequence of applying the isoperimetric inequality to the function’s level sets. Let f be a smooth function that passes through zero at least once on a sufficiently nice (convex, bounded, lipschitz) domain Ω which has radius h.
The 1-norm of f can be approximated by slicing it into slabs along it’s level sets. The integral of |f| is approximately equal to the sum of the volume of the slabs.
The integral of ||∇|| can be broken up into the sum of integrals over the regions between adjacent level set slices.
Thus the relationship between a function and it’s gradient is really the relationship between the areas and perimeters of its level sets.
However, we know how to relate perimeters to areas for a circle, and we know that a circle is the shape that maximizes the perimeter for a given area. thus we can bound the area sum by a weighted perimeter sum.
Finally, the weighting factors that depend on the square root of the areas can be bounded by size of the domain, yielding the well-known Poincare inequality.
* Relating the derivative to the perimeter can be seen more rigorously by considering the distributional derivative of the level set’s indicator function – you end out with a “delta function curve” that tracks the boundary.
* Thanks to Lexing Ying for a short discussion we had. His idea about how to apply the isoperimetric inequality greatly simplified a section of this derivation.