Cañete–Miranda Jr–Vittone planar strip isoperimetric conjecture

Let $S$ be a horizontal strip in $\mathbb{R}^2$ , and let the density be $\rho(x)=1$ on $S$ and $\rho(x)=\lambda>1$ on $\mathbb{R}^2\setminus S$ . For every $\lambda>1$ and every admissible four-arc region $E$ , there should exist an admissible three-arc region $F$ such that $\int_F\rho\,dx=\int_E\rho\,dx$ and $P_\rho(F)<P_\rho(E)$ , where $P_\rho$ denotes weighted perimeter. Equivalently, no four-arc region can be isoperimetric when a three-arc competitor with the same weighted area is available.

References

Primary source

arXiv

Additional references

A comparison theorem for the planar strip isoperimetric problem — arXiv — Skilyn Leon, Rosa Pavlak, Evelyn Pulla, Xi Sisi Shen

geometric-analysis isoperimetric-inequalities

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A new unrefereed preprint claims to settle the conjecture for every allowed density contrast, but the claim has not been independently verified.

Cañete, Miranda Jr., and Vittone posed the conjecture in 2009 for a planar strip whose density is $1$ inside the strip and $\lambda>1$ outside it.

Known results

Cañete, Miranda Jr., and Vittone (2009): the first candidate family is optimal for $v\leq\pi$ .
For $\pi<v\leq v_0$ , the second candidate family is optimal, with $v_0>\pi$ .
For $v_0\leq v\leq v_1$ , the third or fourth candidate family is optimal; for $v\geq v_1$ , the third is optimal.
The fourth family never occurs when $\lambda\geq4/\pi$ , settling the conjectured classification in that range.

September 2026 claimed resolution

Skilyn Leon, Rosa Pavlak, Evelyn Pulla, and Xi Sisi Shen’s preprint A comparison theorem for the planar strip isoperimetric problem claims a reduction valid for every $\lambda>1$ , with an explicit positive perimeter gap, and therefore claims the full conjecture is settled. This is a newly posted, unrefereed claim with no independent verification reported in the retrieved sources.

Current status (as of September 2026): The conjecture is classically settled for $\lambda\geq4/\pi$ and claimed settled for all $\lambda>1$ by a new unverified preprint; the remaining general claim is not independently confirmed.

Sources

No solutions have been posted yet.