Lecture III
We proved in Lecture II that if f r is defined by $${f_r}\left( x \right) = {\left( {\sum\limits_{j = 1}^n {{{\left| {{x_j}} \right|}^r}} } \right)^{1/r}},{\kern 1pt} {\kern 1pt} x \in {\mathbb{R}^n}{\kern 1pt} ,{\kern 1pt} r \geqslant 1{\kern 1pt} ,$$ (1
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Carl Ludwig Siegel Lectures on the
Geometry of Numbers Notes by B. Friedman Rewritten by
Komaravolu Chandrasekharan with the Assistance of Rudolf Suter
With 35 Figures
Springer-V erlag Berlin Heidelberg GmbH
Komaravolu Chandrasekharan Mathematik, ETH Ziirich CH-8092 Ziirich, Switzerland
Mathematics Subject Classification (1980): OI-XX, Il-XX, 12-XX, 15-XX, 20-XX, 32-XX, 51-XX, 52-XX
Library of Congress Cataloging-in-Publication Data. Siegel, Cari Ludwig, 1896-1981 Lccturcs on the geometry of numbers / Cari Ludwig Siegel; notes by B. friedman; rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter. p. cm. Bibliography: p. Includes index. (U.S. : alk. paper) ISBN 978-3-642-08076-0 ISBN 978-3-662-08287-4 (eBook) DOI 10.1007/978-3-662-08287-4 1. Geometry of numbers. 1. Chandrasekharan, K. (Komaravolu), 1920 -. II. Suter, RudolC 1963 -. III. Title. QA241.5.S54 1989512'.5 - dcl9 89-5946 CIP This work is subject ta copyright. AII rights are reserved, whether the whole or pari ofthe material is coneerned, specifically those oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication ofthis publication or parts thereof is only permitted under the provisions ofthe German Copyright Law ofSeptember9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fali under the prosccution act ofthe German Copyright Law. r > 0, we have a > rba,
5
§ 2. Convex hoclies
and therefore R' f= Q. Construct the point P' on the ray emanating from Q and passing through R', so that ~~~:~ = 1· Then P' belongs to the interior of the ball of radius r around P. Therefore P' will be a point in K, and since Q is also in K, we have R' E K. Since R' was an arbitrary point in a ball around R, we have proved that R is an interior point of K.
§ 2. Convex bodies We introduce some definitions for well-known ideas. Definition. A convex body is a bounded, convex, open set in lRn. The interior of an n-dimensional ball, defined by 2 2 x 21 +x 22 + ... +xn 0. Because of (1) and (2), we then have f(J-Lx)
oB,
= f(J-LAY) = J-LA = J-Lf(x).
We note the trivial Theorem 5. If f is the gauge function of a convex body B C IRn containing the origin 0, x E IRn, then f(x) > 0 for x f:. 0, while f(O) = 0.
Note that the properties of the gauge function J, as expressed in Theorems 4 and 5, are also properties of the distance function I I, which assigns to a vector x E IRn (representing the point X) the distance of X from the origin, that is lxl = IOXI = (xi + ... +x;.) 112 , where x = (x1, ... ,xn)· The distance function is the gauge function of the n-dimensional unit ball; it has, however, a third very important property, namely it satisfies the triangle inequality. We shall show that an arbitrary gauge function also has this property. Theorem 6. If f is the gauge function of a convex body B C IRn containing the origin 0, and x, y E IRn, then
f(x
+ y)::;
f(x)
+ f(y).
[This, tagether with the property ex
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