152:, 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.
171:
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
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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
144:). 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
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184:. 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.
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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
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Berichte ĂĽber die
Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig
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Additional examples can be obtained by combining multiple bodies of constant brightness using the
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The first known body of constant brightness that is not a sphere was constructed by
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Howard, Ralph (2006), "Convex bodies of constant width and constant brightness",
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Bodies of
Constant Width: An Introduction to Convex Geometry with Applications
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260:(1915), "Einige Bemerkungen über Kurven und Flächen von konstanter Breite",
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66:, all bodies of constant brightness that have the same projected area
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The name comes from interpreting the body as a shining body with
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214:(2019), "Section 13.3.2 Convex Bodies of Constant Brightness",
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Gronchi, Paolo (1998), "Bodies of constant brightness",
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Jahresbericht der
Deutschen Mathematiker-Vereinigung
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at pairs of opposite points of tangency of parallel
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51:A body has constant brightness if and only if the
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329:"Eine charakteristische Eigenschaft der Kugel"
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113:{\displaystyle \textstyle {\sqrt {A/\pi }}}
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210:Martini, Horst; Montejano, Luis;
218:, Birkhäuser, pp. 310–313,
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53:reciprocal Gaussian curvatures
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120:. This can be proved by the
62:According to an analogue of
136:in 1915. Its boundary is a
22:body of constant brightness
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174:surfaces of constant width
163:Relation to constant width
34:surfaces of constant width
377:10.1016/j.aim.2005.05.015
224:10.1007/978-3-030-03868-7
405:Euclidean solid geometry
30:curves of constant width
353:Advances in Mathematics
274:2027/mdp.39015036849837
169:curve of constant width
24:is a three-dimensional
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327:Nakajima, S. (1926),
306:10.1007/s000130050224
292:Archiv der Mathematik
138:surface of revolution
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41:isotropic luminance
182:Nakajima's problem
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258:Blaschke, Wilhelm
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212:Oliveros, DĂ©borah
142:Reuleaux triangle
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79:{\displaystyle A}
64:Barbier's theorem
57:supporting planes
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410:Geometric shapes
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268:: 290–297,
399:Categories
188:References
47:Properties
26:convex set
339:: 298–300
105:π
150:football
386:2233133
315:1622002
242:3930585
128:Example
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362:arXiv
228:ISBN
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