Bieberbach's inequality

The Bieberbachsche inequality is a result of the convex geometry , which after the mathematician Ludwig Bieberbach is named (1886-1982). It deals with the connection between volume and diameter of certain distinguished subsets of n-dimensional Euclidean space .

The inequality

The Bieberbachsche inequality can be formulated as follows:

For a non-empty compact convex body of -dimensional Euclidean space, the inequality always applies with regard to its -dimensional volume and its diameter ${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle V(A)}$ ${\displaystyle diam(A)}$

${\displaystyle V(A)\leq {\frac {V_{n}}{2^{n}}}\cdot (diam(A))^{n}}$

where the volume of the -dimensional unit sphere means. ${\displaystyle V_{n}}$ ${\displaystyle n}$

In this inequality, equality exists if and only if it coincides with a -dimensional sphere . ${\displaystyle A}$ ${\displaystyle n}$

Development history

In 1915 Ludwig Bieberbach demonstrated the inequality named after him for the Euclidean plane . It was then generalized by various authors and initially transferred to three-dimensional space by Wilhelm Blaschke . This was followed by the further generalization of the inequality to Euclidean spaces of higher dimensions and then even to non-Euclidean spaces . Above all, Erhard Schmidt and some Russian mathematicians such as Paul Urysohn played a major role in this further development . As can be shown, Bieberbach's inequality arises particularly as a consequence of a general inequality over mixed volumes from Alexandroff - Fennel .

See also

• Isoperimetric inequality

literature

• Ludwig Bieberbach : About an extremal property of the circle . In: Jber. Dtsch. Math.-Ver . tape 24 , 1915, pp. 247-250 ( uni-goettingen.de ).
• Yu. D. Burago , VA Zalgaller : Geometric Inequalities (=  The basic teachings of the mathematical sciences in single representations . Volume 285 ). Springer Verlag, Berlin ( inter alia ) 1988, ISBN 3-540-13615-0 ( MR0936419 ).
• Wilhelm Blaschke : circle and sphere . Chelsea Publishing Company, New York ( inter alia ) 1949 ( MR0076364 MR0077958 - reprint of the edition by Veit [Leipzig 1916]).
• H. Hadwiger : Lectures on content, surface and isoperimetry (=  the basic teachings of the mathematical sciences in individual presentations with special consideration of the areas of application . Volume 93 ). Springer-Verlag, Berlin ( inter alia) 1957 ( MR0102775 ).
• Erhard Schmidt: The Brunn-Minkowski theorem and its mirror theorem as well as the isoperimetric property of the sphere in Euclidean and hyperbolic geometry . In: Math. Ann . tape 120 , 1948, pp. 307-422 ( MR0028601 uni-goettingen.de ).
• Erhard Schmidt: The Brunn-Minkowski inequality and its mirror image as well as the isoperimetric property of the sphere in Euclidean and non-Euclidean geometry I, II . In: Mathematical News . tape 1, 2 (1948/1949) , pp. 81-157 (1948), 171-244 (1949) ( MR0028600 MR0034044 ).
• Paul Urysohn: Mean width and volume of the convex bodies in n-dimensional space . In: Matem. Sb.SSSR . tape 31 , 1924, pp. 477-486 .

References and footnotes

1. ↑ Burago-Zalgaller: p. 93.
2. ↑ Hadwiger, p. 173.
3. ↑ Hugo Hadwiger also calls such a body an egg body ; see. Hadwiger, p. 198.
4. ↑ The n-dimensional volume or - in the two-dimensional case - the area of ​​an egg body agrees with its Lebesgue measure ; see. Hadwiger, p. 157.
5. ↑ Bieber Bach: .. Jber dtsch Math.-Ver. No. 24 , p. 247 ff .
6. ↑ Blaschke, p. 122 ff.
7. ↑ Burago-Zalgaller: pp. 93 ff, 143 ff.
8. ↑ Hadwiger, pp. 178-179.
9. ↑ Schmidt: Math. Nachr. No. 1/2 , p. 81 ff., 171 ff .