September 29, 2025

Bounded Variation Separates Weak and Strong Average Lipschitz.

Puntos clave

Weak average-Lipschitz is shown to be weaker than bounded variation, while strong average-Lipschitz is stronger.
The relationship highlights that the weak average smooth class is significantly larger in combinatorial terms due to the fat-shattering dimension.
Average Lipschitz seminorms extend to real-valued functions on metric spaces, enhancing understanding of smoothness.
Insights may impact various applications, particularly in mathematical analysis and optimization.

Resumen

We closely examine a recently introduced notion of average smoothness. The latter defined a weak and strong average-Lipschitz seminorm for real-valued functions on general metric spaces. Specializing to the standard metric on the real line, we compare these notions to bounded variation (BV) and discover that the weak notion is strictly weaker than BV while the strong notion is strictly stronger. Along the way, we discover that the weak average smooth class is also considerably larger in a certain combinatorial sense, which is made precise by the fat-shattering dimension.

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