A body has constant brightness if and only if the reciprocal Gaussian curvatures at pairs of opposite points of tangency of parallel supporting planes have almost-everywhere-equal sums.
[1][2] According to an analogue of Barbier's theorem, all bodies of constant brightness that have the same projected area
[3] Additional examples can be obtained by combining multiple bodies of constant brightness using the Blaschke sum, an operation on convex bodies that preserves the property of having constant brightness.
[3] A curve of constant width in the Euclidean plane has an analogous property: all of its one-dimensional projections have equal length.
Nakajima himself proved the conjecture under the additional assumption that the boundary of the shape is smooth.