Submitted Paper
Inserted: 7 nov 2026
Last Updated: 7 nov 2026
Year: 2026
Abstract:
Talenti symmetrization rearranges functions on an open connected set \(X\subset\mathbb{R}^N\), endowed with a measure \(\mu\) with continuous positive density, along the superlevel sets of a given function \(f\). Talenti proved a Pólya–Szegő principle for it, provided that these sets are \(\mu\)-isoperimetric, have connected boundaries, and are crossed by straight lines of steepest ascent. We show that, if \(X\) is convex, these requirements are very rigid: they force the superlevel sets of \(f\) to be either parallel half-spaces, or concentric balls, or complements of tubes around an affine subspace of codimension at least two. This identifies the only geometries in which Talenti's Pólya–Szegő principle can be applied. We also prove a family of weighted isoperimetric inequalities in convex cones, showing that complements of tubes around affine subspaces of every codimension at least two do occur.
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