Banach's Indicatrix Reloaded

Abstract

Banach famously related the smoothness of a function to the size of its level sets. More precisely, he showed that a continuous function is of bounded variation exactly when its "indicatrix" is integrable. In a similar vein, we connect the smoothness of the function -- measured now by its integral modulus of continuity -- to the structure of its superlevel sets. Our approach ultimately reduces to a continuum incidence problem for quantifying the regularity of open sets. The pay off is a refinement of Banach's original theorem and an answer to a question of Garsia--Sawyer.