Convex-body surface-area capacity conjecture

Prove that among convex sets in three-dimensional space with a prescribed surface area, the degenerate double-sided disk minimizes Newtonian capacity.

Background

The paper identifies a separate Pólya–Szegő capacity conjecture concerning convex bodies in three-dimensional space. Unlike the main conjecture, this problem imposes a surface-area constraint and seeks the minimum Newtonian capacity.

The proposed extremizer is the degenerate double-sided disk. The paper notes that partial results are known and mentions the possible relevance of Jerison’s solution of the Minkowski problem for Newtonian capacity.

References

Another well known capacity conjecture by Pólya and Szegő that still remains open is the claim that among convex sets in 3-space with given surface area, the Newtonian capacity \capone is minimal for the degenerate double-sided disk.

— Toward Pólya and Szegő's conjecture for logarithmic vs Newtonian capacity  (2609.35438 - Clark et al., 28 Sep 2026) in Section 1, subsection “Related problems and literature”