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Martin Sleziak
Martin Sleziak
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I'm pretty sure that this is still open for $n$-gons (with $n\geq 5$). As far as I know, basically no progress has been made since the original proofs for triangles/quadrilaterals.

There have been some numerics as well as some refined inequalities for triangles. This articleThis article might point you towards some of these results.

Interestingly, a related conjecture of Pólya & Szegő is resolved "the regular $n$-gon has least logarithmic capacity among $n$-gons of a fixed area" http://www.ams.org/mathscinet-getitem?mr=2052355.

I'm pretty sure that this is still open for $n$-gons (with $n\geq 5$). As far as I know, basically no progress has been made since the original proofs for triangles/quadrilaterals.

There have been some numerics as well as some refined inequalities for triangles. This article might point you towards some of these results.

Interestingly, a related conjecture of Pólya & Szegő is resolved "the regular $n$-gon has least logarithmic capacity among $n$-gons of a fixed area" http://www.ams.org/mathscinet-getitem?mr=2052355.

I'm pretty sure that this is still open for $n$-gons (with $n\geq 5$). As far as I know, basically no progress has been made since the original proofs for triangles/quadrilaterals.

There have been some numerics as well as some refined inequalities for triangles. This article might point you towards some of these results.

Interestingly, a related conjecture of Pólya & Szegő is resolved "the regular $n$-gon has least logarithmic capacity among $n$-gons of a fixed area" http://www.ams.org/mathscinet-getitem?mr=2052355.

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Otis Chodosh
Otis Chodosh
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I'm pretty sure that this is still open for $n$-gons (with $n\geq 5$). As far as I know, basically no progress has been made since the original proofs for triangles/quadrilaterals.

There have been some numerics as well as some refined inequalities for triangles. This article might point you towards some of these results.

Interestingly, a related conjecture of Pólya & Szegő is resolved "the regular $n$-gon has least logarithmic capacity among $n$-gons of a fixed area" http://www.ams.org/mathscinet-getitem?mr=2052355.