Clark–Laugesen capacity-ratio conjectures
Let denote the Riesz -capacity of a compact set . The Clark–Laugesen conjectures ask for sharp extremal inequalities for the ratio when , including the conjectured extremality of the Euclidean ball over a broad interval of exponents and the corresponding two-point extremal conjecture. The supplied sources do not state the precise endpoint ranges, admissible class of sets, or inequality directions sufficiently explicitly to give a more detailed formal statement without adding unsupported assumptions.
References
Primary source
Additional references
- Riesz capacity ratios with negative exponents — arXiv — Qiuling Fan
Progress summary
A September 2026 paper reports new tests and partial results for several capacity-ratio conjectures, but the main broad and two-point conjectures remain open.
The entry concerns several Clark–Laugesen conjectures about extremal capacity ratios. Publicly retrieved material does not report a complete proof or counterexample for the broad interval or two-point conjectures.
September 2026 capacity-ratio evidence
On September 10, 2026, a summary of Qiuling Fan’s paper Riesz capacity ratios with negative exponents reported tests and partial proofs for named extremal-ratio conjectures, including a large-dimensional regular-simplex comparison for negative exponents. The broad interval and two-point conjectures were not claimed solved. This is claimed progress, not independently verified here.
March 2026 related conjecture
A 2026 paper proposed a strip-energy conjecture implying related capacity comparisons, including the planar Pólya–Szegő conjecture, and stated that the original conjecture remains open; it supplied no proof or counterexample.
Current status (as of September 2026): Several parameter-specific comparisons have claimed partial progress, but the broad interval and two-point Clark–Laugesen conjectures remain open.
Solutions 0
No solutions have been posted yet.