mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Shephard's problem, is the following geometrical question asked by

Geoffrey Colin Shephard Geoffrey Colin Shephard is a mathematician who works on convex geometry and reflection groups. He asked Shephard's problem on the volumes of projected convex bodies, posed another problem on polyhedral nets, proved the Shephard–Todd theorem in ...

in 1964: if ''K'' and ''L'' are centrally symmetric

convex bodies In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...

in ''n''-

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

such that whenever ''K'' and ''L'' are

projected Projected is an American rock supergroup consisting of Sevendust members John Connolly and Vinnie Hornsby, Alter Bridge and Creed drummer Scott Phillips, and former Submersed and current Tremonti guitarist Eric Friedman. The band released thei ...

onto a

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

, the

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

of the projection of ''K'' is smaller than the volume of the projection of ''L'', then does it follow that the volume of ''K'' is smaller than that of ''L''? In this case, "centrally symmetric" means that the

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in s ...

of ''K'' in the origin, ''−K'', is a translate of ''K'', and similarly for ''L''. If _''k'' : R^''n'' → Π_''k'' is a

projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...

of R^''n'' onto some ''k''-dimensional

Π_''k'' (not necessarily a coordinate hyperplane) and ''V''_''k'' denotes ''k''-dimensional volume, Shephard's problem is to determine the truth or falsity of the implication :

V_ (\pi_ (K)) \leq V_ (\pi_ (L)) \mbox 1 \leq k < n \implies V_ (K) \leq V_ (L).

''V''_''k''(_''k''(''K'')) is sometimes known as the brightness of ''K'' and the function ''V''_''k'' o _''k'' as a (''k''-dimensional) brightness function. In dimensions ''n'' = 1 and 2, the answer to Shephard's problem is "yes". In 1967, however, Petty and Schneider showed that the answer is "no" for every ''n'' ≥ 3. The solution of Shephard's problem requires

Minkowski's first inequality for convex bodies In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn–Minkowski inequality and the isoperimetric inequality. State ...

and the notion of projection bodies of convex bodies.

Notes

References

* * * *{{Citation , last1=Shephard , first1=G. C. , title=Shadow systems of convex sets , doi=10.1007/BF02759738 , doi-access=free , mr=0179686 , year=1964 , journal=

Israel Journal of Mathematics '' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the jou ...

, issn=0021-2172 , volume=2 , issue=4 , pages=229–236 Convex geometry Convex analysis

See also

Notes

References