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Pansu Conjecture

The Pansu conjecture asserts that the Pansu balls minimize horizontal perimeter among measurable sets of fixed volume in the first Heisenberg group.

Chen et al. (2026) reported the unrestricted sharp isoperimetric inequality

P_H(E)^4>=(512pi^2)/(27)V(E)^3

for 0<V(E)<infty and finite horizontal perimeter P_H(E), with equality iff E agrees with a Pansu ball except on a set of measure zero. Here V is Lebesgue measure in the coordinates and normalization used in the definition of a Pansu ball.

See also

Heisenberg Group, Horizontal Perimeter, Isoperimetric Inequality, Pansu Ball

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References

Chen, W.-C.; Kao, W.-T.; and Lien, C.-K. "The Sharp Isoperimetric Inequality in the First Heisenberg Group." 6 Oct 2026. https://arxiv.org/abs/2610.07632.

Cite this as:

Weisstein, Eric W. "Pansu Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PansuConjecture.html

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