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Pansu Ball

A Pansu ball is a translate and dilate of the following region in the first Heisenberg group. In the normalization with the group operation

(x,y,t)(x^',y^',t^')=(x+x^',y+y^',t+t^'+yx^'-xy^'),
(1)

the unit Pansu ball is

Omega={(z,t):|z|<1, |t|<g(|z|)},
(2)

where z=x+iy and

g(r)=1/2(cos^(-1)r+rsqrt(1-r^2)).
(3)

The other Pansu balls are its left translates and its dilates under delta_s(z,t)=(sz,s^2t), s>0. With Lebesgue measure as volume and the horizontal perimeter associated with X=partial_x+ypartial_t and Y=partial_y-xpartial_t, the unit Pansu ball has volume 3pi^2/8 and horizontal perimeter pi^2 (Chen et al. 2026).

See also

Heisenberg Group, Horizontal Perimeter, Pansu Conjecture

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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 Ball." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PansuBall.html

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