For perimetral density σ=1/6, the disk is the only convex domain that floats in equilibrium in every position.
Dupin, editor.Applications de g´ eom´ etrie et de m´ echanique ` a la marine, aux ponts-et- chauss´ ees
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Rigidity in the Planar Ulam Floating Body Problem with perimetral density $\sigma=\tfrac16$
For perimetral density σ=1/6, the disk is the only convex domain that floats in equilibrium in every position.