convex geometry In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbe ...

, a body of constant brightness is a three-dimensional

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...

all of whose two-dimensional projections have equal area. A sphere is a body of constant brightness, but others exist. Bodies of constant brightness are a generalization of

curves of constant width In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width ...

, but are not the same as another generalization, the surfaces of constant width. The name comes from interpreting the body as a shining body with isotropic luminance, then a photo (with focus at infinity) of the body taken from any angle would have the same total light energy hitting the photo.

Properties

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. According to an analogue of

Barbier's theorem In geometry, Barbier's theorem states that every curve of constant width has perimeter times its width, regardless of its precise shape. This theorem was first published by Joseph-Émile Barbier in 1860. Examples The most familiar examples of c ...

, all bodies of constant brightness that have the same projected area

A

as each other also have the same surface area,

\textstyle\sqrt

. This can be proved by the

Crofton formula In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it. Statement Suppose \gamma is a ...

Example

The first known body of constant brightness that is not a sphere was constructed by

Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taught ...

in 1915. Its boundary is a

surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...

of a curved triangle (but not the Reuleaux triangle). It is smooth except on a circle and at one isolated point where it is crossed by the axis of revolution. The circle separates two patches of different geometry from each other: one of these two patches is a

spherical cap In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere (forming a ...

, and the other forms part of a

football Football is a family of team sports that involve, to varying degrees, kicking a ball to score a goal. Unqualified, the word ''football'' normally means the form of football that is the most popular where the word is used. Sports commonly c ...

, a surface of constant Gaussian curvature with a pointed tip. Pairs of parallel supporting planes to this body have one plane tangent to a singular point (with reciprocal curvature zero) and the other tangent to the one of these two patches, which both have the same curvature. Among bodies of revolution of constant brightness, Blaschke's shape (also called the Blaschke–Firey body) is the one with minimum volume, and the sphere is the one with maximum volume. Additional examples can be obtained by combining multiple bodies of constant brightness using the

Blaschke sum In convex geometry and the geometry of convex polytopes, the Blaschke sum of two polytopes is a polytope that has a facet parallel to each facet of the two given polytopes, with the same measure. When both polytopes have parallel facets, the meas ...

, an operation on convex bodies that preserves the property of having constant brightness.

Relation to constant width

curve of constant width In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or ...

in the Euclidean plane has an analogous property: all of its one-dimensional projections have equal length. In this sense, the bodies of constant brightness are a three-dimensional generalization of this two-dimensional concept, different from the surfaces of constant width. Since the work of Blaschke, it has been conjectured that the only shape that has both constant brightness and constant width is a sphere. This was formulated explicitly by Nakajima in 1926, and it came to be known as ''Nakajima's problem''. Nakajima himself proved the conjecture under the additional assumption that the boundary of the shape is smooth. A proof of the full conjecture was published in 2006 by Ralph Howard.

References

{{reflist, refs= {{citation , last = Blaschke , first = Wilhelm , author-link = Wilhelm Blaschke , hdl = 2027/mdp.39015036849837 , journal = Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig , pages = 290–297 , title = Einige Bemerkungen über Kurven und Flächen von konstanter Breite , volume = 67 , year = 1915 {{citation , last = Gronchi , first = Paolo , doi = 10.1007/s000130050224 , doi-access=free , issue = 6 , journal =

Archiv der Mathematik '' Archiv der Mathematik'' is a peer-reviewed mathematics journal published by Springer, established in 1948. Abstracting and indexing The journal is abstracted and indexed in:

, mr = 1622002 , pages = 489–498 , title = Bodies of constant brightness , volume = 70 , year = 1998 {{citation , last = Howard , first = Ralph , doi = 10.1016/j.aim.2005.05.015 , doi-access=free , issue = 1 , journal =

Advances in Mathematics ''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed ...

, mr = 2233133 , pages = 241–261 , title = Convex bodies of constant width and constant brightness , volume = 204 , year = 2006, arxiv = math/0306437 {{citation , last1 = Martini , first1 = Horst , last2 = Montejano , first2 = Luis , last3 = Oliveros , first3 = Déborah , author3-link = Déborah Oliveros , contribution = Section 13.3.2 Convex Bodies of Constant Brightness , doi = 10.1007/978-3-030-03868-7 , isbn = 978-3-030-03866-3 , mr = 3930585 , pages = 310–313 , publisher = Birkhäuser , title = Bodies of Constant Width: An Introduction to Convex Geometry with Applications , year = 2019 {{citation , last = Nakajima , first = S. , journal = Jahresbericht der Deutschen Mathematiker-Vereinigung , pages = 298–300 , title = Eine charakteristische Eigenschaft der Kugel , url = https://eudml.org/doc/145745 , volume = 35 , year = 1926 Euclidean solid geometry Geometric shapes Constant width