TheOne generalizationway ofto thegeneralize definitionthese of bodies of constant widthconcepts to convexthree bodiesdimensions inis <math>\mathbb{R}^3</math> and their boundaries leads tothrough the concept of [[surface of constant width|surfaces of constant width]]. (inThe thethree-dimensional caseanalog of a Reuleaux triangle, this does not lead to athe [[Reuleaux tetrahedron]], does not have constant width, but tosmall modifications of it, the [[Meissner bodies]]), do.{{r|gardner|mmo}} ThereA isdifferent alsoclass aof conceptthree-dimensional ofgeneralizations, the [[space curve]]s of constant width, are defined by the properties that each plane that crosses the curve perpendicularly intersects it at exactly one other point, where it is also perpendicular, and that all pairs of points intersected by perpendicular planes are the same distance apart.{{r|fujiwara|cieslak|teufel|wegner72}}
Curves and bodies of constant width have also been studied in [[non-Euclidean geometry]]{{r|leichtweiss}} and for non-Euclidean [[normed vector space]]s.{{r|eggleston}}
