Polynomial threshold functions (PTFs) form an important low-complexity class of Boolean functions, with deep connections to learning theory and approximation theory. Recent progress on learning and testing PTFs has relied heavily on their structural and isoperimetric properties, particularly on bounds for average sensitivity, a central theme since the Gotsman--Linial conjecture.
In this work, we reveal a new geometric constraint governing PTFs by analyzing them through the lens of the \emph{Boolean surface area} (also known as the Talagrand boundary):
\[
\mathrm{BSA}[f] = \mathbb{E}\,|\nabla f| = \mathbb{E}\sqrt{\mathrm{Sens}_f(x)},
\]
which serves as a natural measure of vertex-boundary complexity on the discrete cube. Our main result shows that every degree-$d$ PTF $f$ has subpolynomial Boolean surface area:
\[
\mathrm{BSA}[f] \le \exp\!\big(C(d)\sqrt{\log n}\big).
\]
This bound represents a superpolynomial improvement over the previous estimate
\[
n^{1/4} (\log n)^{C(d)},
\]
which follows from Kane's landmark results on the average sensitivity of PTFs.
Consequently, degree-$d$ PTFs satisfy a stronger form of geometric regularity than was previously visible from influence bounds alone. As an application, we obtain improved noise sensitivity estimates in the regime of small noise parameters.
