Differentiability of the volume of a region enclosed by level sets

TL;DR

This paper investigates the smoothness properties of volume functions defined by level sets of smooth functions, showing they are smooth at regular values and finitely differentiable at critical values, with applications in radiotherapy optimization.

Contribution

It establishes the differentiability properties of volume functions related to level sets, highlighting their behavior at regular and critical values, relevant for optimization tasks.

Findings

Volume functions are smooth at regular values.

Volume functions are finitely differentiable at critical values.

Applications in radiotherapy optimization are impacted by these properties.

Abstract

The level of a function f on an n-dimensional space encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the function f is smooth, the volume function is again smooth for regular values of f. For critical values of f the volume function is only finitely differentiable. The initial motivation for this study comes from Radiotherapy, where such volume functions are used in an optimization process. Thus their differentiability properties become important.