Author Index Achatz, 277 Agarwal, 354, 450, 500 Agrachev, 64 Aharonov, 223 Ahmedov, 408 Aichholzer, 239 Aigner, 289, 293 Alesker, 110, 133 Alex<strong>and</strong>roff, 43, 257, 259, 314 Alex<strong>and</strong>rov, 28, 40, 79, 102, 125, 188, 193, 197, 198, 244, 292, 297, 301–304, 470, 471 Alfsen, 34 Aliev, 321, 389, 391 Almgren, 160 Alon, 279 Alt, 239 Ambartzumian, 135 Amir, 226, 229 Anderson, 26, 69 Andreev, 500 Andrews, 200 Andriyasyan, 407 Archimedes, 41 Arias de Reyna, 167 Aristotle, 80 Arnold, 201 Artin, 17 Artstein, 61 Asimow, 292, 297, 299 Aurenhammer, 464 Ausl<strong>and</strong>er, 299 Babai, 419 Baire, 71 Balinski, 274 Ball, 39, 40, 124, 141, 148, 164, 167, 175, 203, 204, 208, 358, 426 Bambah, 376, 405–408, 421, 430, 440, 462 567 B<strong>and</strong>le, 168, 178, 179, 184 B<strong>and</strong>t, 234, 235 Bangert, 231 Baraki, 234, 235 Baranovskiĭ, 354 Bárány, 203, 217, 512 Barnes, 435, 439 Barnette, 270 Bartels, 336 Barthe, 141, 208 Barvinok, 251, 310, 316 Barykinskiĭ, 354 Bauer, 191, 223, 225 Bayer, 244, 259, 265, 271, 272, 278 Bayer-Fluckiger, 405 Beck, 311 Beckenbach, 1 Beer, 88 Behrend, 203, 204, 208 Belhoste, 293 Bender, 278, 279 Beneˇs, 135 Benson, 154 Benyamini, 26, 40 Berger, 201 Berkovitz, 336 Bernstein, 320, 333, 335 Bertr<strong>and</strong>, 375 Bertraneu, 431 Bertsimas, 349 Betke, 110, 257, 324, 325, 448 Bezdek, A., 450, 454 Bezdek, K., 426 Bianchi, 174 Bieberbach, 152, 175 Bigalke, 222 Billera, 265, 269 Birch, 406, 407 Birkhoff, G., 77, 229 Birkhoff, G.D., 201
568 AUTHOR INDEX Björner, 265 Blaschke, 40, 84, 109, 110, 132, 135, 142, 172, 174–176, 226, 229 Blichfeldt, 369, 376, 422, 424 Blind, 275 Blömer, 375 Böhm, 280 Bohr, 17, 18 Bokowski, 354, 457 Bol, 88, 101 Bol’shakova, 470, 471 Bolker, 64 Bollobás, 262, 501 Boltyanskiĭ, 46, 64, 289 Bolyai, 288, 289 Bombieri, 354, 407 Bonnesen, 40, 222, 226 Boots, 464 Borgwardt, 336 Böröczky, Jr., 100, 160, 187, 215, 354, 445, 491 Borwein, 1, 52 Bourgain, 124, 174, 177 Brascamp, 161, 208 Brass, 354 Brechtken–M<strong>and</strong>erscheid, 36 Bricard, 288, 292, 297 Brightwell, 500, 501 Brion, 311 Brøndsted, 244, 259, 265 Bruggesser, 265 Brunn, 1, 32, 79, 142, 151 Buchta, 203, 218 Buffon, 135 Burago, 40, 94, 102, 148, 164, 222 Burkard, 336 Busemann, 28, 53, 124, 154, 158, 188, 197 Buser, 185 Caffarelli, 196 Campi, 197 Cappell, 410 Carathéodory, 43, 46 Carroll, 321 Cartier, 289 Cassels, 354, 355, 361, 366, 385, 391, 399, 410 Castillo, 13, 17 Cauchy, 79, 106, 292 Chakerian, 185 Chalk, 408 Chavel, 148 Chebotarev, 407 Chen, 126, 129, 375 Chern, 135 Chernaya, 480, 484, 498, 499 Chiba, 262 Chiu, 464 Choquet, 187, 219 Chou, 200 Cohn, 422, 429 Cole, 408 Colesanti, 70, 197 Colin de Verdière, 500, 509 Collins, 509 Connelly, 292, 293, 297 <strong>Convex</strong>ity <strong>and</strong> Its Applications, 40 Conway, 185, 354, 355, 421, 427, 439 Coppel, 46, 354 Cornfeld, 201 Corput, 349 Corrádi, 480 Courant, 36 Cox, 335 Coxeter, 168, 218, 244, 438, 439, 462, 478 Czerwik, 1, 17 Danilov, 399 Danilova, 149, 169 Dantzig, 336, 339 Danzer, 46, 203, 204, 207, 293 Davenport, 354, 385, 406, 407 Davidoff, 354 Day, 226 Debrunner, 46 Dehn, 265, 269, 289, 292 DeLoera, 310, 311, 316
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Grundlehren der mathematischen Wiss
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Peter M. Gruber Institute of Discre
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VI Preface theory of finite groups
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Contents Preface ..................
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Contents XI 12 Special Convex Bodie
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Contents XIII 29 Packing of Balls a
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2 Convex Functions 1 Convex Functio
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4 Convex Functions x epi f f C y Fi
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6 Convex Functions Together these i
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8 Convex Functions Theorem 1.4. Let
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10 Convex Functions z = f (x) + f
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12 Convex Functions 1.3 Convexity C
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14 Convex Functions Proof (by affin
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16 Convex Functions Minkowski’s I
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18 Convex Functions (ii) Let x > 0.
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20 Convex Functions 2 Convex Functi
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22 Convex Functions | f (z) − f (
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24 Convex Functions First-Order Dif
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26 Convex Functions Remark. More pr
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28 Convex Functions even more is tr
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30 Convex Functions G(x). The proof
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32 Convex Functions by (15), unifor
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34 Convex Functions 2.4 A Stone-Wei
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36 Convex Functions x x Fig. 2.1. S
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38 Convex Functions F is convex. Us
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40 Convex Bodies results, character
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42 Convex Bodies Convex Hulls Given
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44 Convex Bodies The next result is
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46 Convex Bodies C ∩ L ⊥ clearl
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48 Convex Bodies Suppose not. Then,
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50 Convex Bodies The Centrepoint of
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52 Convex Bodies Since G is open, w
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54 Convex Bodies in one of the two
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56 Convex Bodies or geometric infor
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58 Convex Bodies v HC (u, h(u)) (v,
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60 Convex Bodies C D Fig. 4.3. Sepa
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62 Convex Bodies For the proof of t
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64 Convex Bodies in contradiction t
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66 Convex Bodies (6) R(t) ⊆ E d i
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68 Convex Bodies 5 The Boundary of
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70 Convex Bodies for all y ∈ C an
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72 Convex Bodies the existence of s
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74 Convex Bodies where r(·) is a s
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76 Convex Bodies Hence x ∈ conv e
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78 Convex Bodies N = � � X, O Y
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80 Convex Bodies Further topics of
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82 Convex Bodies We are now ready t
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84 Convex Bodies similarly, D j is
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86 Convex Bodies For the proof of t
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88 Convex Bodies The definition of
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90 Convex Bodies If ei ∈ Pi is no
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92 Convex Bodies Let v(·) denote (
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94 Convex Bodies The present sectio
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96 Convex Bodies Proposition 6.5. L
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98 Convex Bodies Thus Second, Minko
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100 Convex Bodies The Valuation Pro
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102 Convex Bodies Minkowski’s the
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104 Convex Bodies To remedy the sit
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106 Convex Bodies Proof. Applying S
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108 Convex Bodies Third, let C be a
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110 Convex Bodies A Problem of Sant
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112 Convex Bodies sets of S. Ifφ c
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114 Convex Bodies = � (−1) |J|
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116 Convex Bodies If P has dimensio
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118 Convex Bodies Since, by Volland
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120 Convex Bodies that is, V is sim
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122 Convex Bodies Let C, D ∈ C su
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124 Convex Bodies P Q . . . . Fig.
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126 Convex Bodies 7.3 Characterizat
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128 Convex Bodies and the Bi have p
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130 Convex Bodies Since the Riemann
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132 Convex Bodies Seventh, o v3 v2
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134 Convex Bodies Mean Width and th
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136 Convex Bodies � χ(C ∩ mD)
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138 Convex Bodies In the following,
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140 Convex Bodies 7.5 Hadwiger’s
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142 Convex Bodies 8.1 The Classical
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144 Convex Bodies C ∩ H(tC ) C D
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146 Convex Bodies Theorem 8.4. Let
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148 Convex Bodies In the following
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150 Convex Bodies Integration impli
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152 Convex Bodies As a consequence
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154 Convex Bodies Proof. � 1 2 w(
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156 Convex Bodies Let ϱ>0 be the m
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158 Convex Bodies Wulff’s Theorem
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160 Convex Bodies Third, by (8) and
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162 Convex Bodies Since the measure
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164 Convex Bodies An application of
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166 Convex Bodies Since f has Lipsc
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168 Convex Bodies Clearly, the inte
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170 Convex Bodies Note that �st :
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172 Convex Bodies Proposition 9.2.
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174 Convex Bodies Proof. Let ε>0.
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176 Convex Bodies The Brunn-Minkows
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178 Convex Bodies 9.3 Schwarz Symme
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180 Convex Bodies Torsional Rigidit
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182 Convex Bodies The proof of (3)
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184 Convex Bodies � �� �2 2
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186 Convex Bodies Theorem 9.10. Let
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188 Convex Bodies Both problems inf
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190 Convex Bodies The Area Measure
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192 Convex Bodies halfspace {z : un
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194 Convex Bodies Let σm be the di
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196 Convex Bodies Thus Minkowski’
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198 Convex Bodies Suppose now that
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200 Convex Bodies vector of St at x
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202 Convex Bodies Outer Billiards F
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204 Convex Bodies For more informat
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206 Convex Bodies with P. Thus, it
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208 Convex Bodies Note that ui ∈
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210 Convex Bodies Theorem 11.4. Let
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212 Convex Bodies the maximum of th
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214 Convex Bodies (20) mni ≥ 1 λ
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216 Convex Bodies Prove analogous r
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218 Convex Bodies Heuristic Princip
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220 Convex Bodies λC + x νC + z
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222 Convex Bodies To show that (K +
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224 Convex Bodies The integral here
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226 Convex Bodies In the following
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228 Convex Bodies The first such re
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230 Convex Bodies v o Fig. 12.3. Sy
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232 Convex Bodies What Does the Bou
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234 Convex Bodies Corollary 13.1. L
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236 Convex Bodies Hence ν = 0 and
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238 Convex Bodies geometry, see Sei
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240 Convex Bodies We distinguish th
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242 Convex Bodies by Proposition 13
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244 Convex Polytopes The material i
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246 Convex Polytopes V-Polytopes an
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248 Convex Polytopes Convex cones w
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250 Convex Polytopes Generalized Co
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252 Convex Polytopes d 2 inequaliti
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254 Convex Polytopes The Face Latti
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256 Convex Polytopes Then (6) impli
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258 Convex Polytopes P in F. For an
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260 Convex Polytopes Euler’s Poly
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262 Convex Polytopes bijection betw
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264 Convex Polytopes The Converse o
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266 Convex Polytopes (geometric) fa
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268 Convex Polytopes consists preci
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270 Convex Polytopes The Lower and
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272 Convex Polytopes A problem of S
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274 Convex Polytopes The first cond
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276 Convex Polytopes The existence
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278 Convex Polytopes For d = 3 it f
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280 Convex Polytopes 16 Volume of P
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282 Convex Polytopes by (2). Hence
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284 Convex Polytopes (12) The volum
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286 Convex Polytopes T ∩ Z, respe
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288 Convex Polytopes An Alternative
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290 Convex Polytopes P clearly can
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292 Convex Polytopes 17 Rigidity Ri
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294 Convex Polytopes Fig. 17.2. Cau
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296 Convex Polytopes Call the index
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298 Convex Polytopes of V. Fori ∈
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300 Convex Polytopes Thus (3) impli
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302 Convex Polytopes Proof. (i)⇒(
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304 Convex Polytopes For other pert
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306 Convex Polytopes where λ is a
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308 Convex Polytopes 18.3 The Isope
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310 Convex Polytopes 19 Lattice Pol
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312 Convex Polytopes Embed E d into
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314 Convex Polytopes To see this, n
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316 Convex Polytopes The Coefficien
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318 Convex Polytopes If the triangl
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320 Convex Polytopes � qP(t) = No
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322 Convex Polytopes Now, take into
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324 Convex Polytopes L � P + (n +
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326 Convex Polytopes V (S) = V .Let
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328 Convex Polytopes (7) ψ|H d = 0
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330 Convex Polytopes Theorem 19.6.
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332 Convex Polytopes (15) L(nP) = L
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334 Convex Polytopes have to be sin
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336 Convex Polytopes economics, von
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338 Convex Polytopes Existence of S
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340 Convex Polytopes Description of
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342 Convex Polytopes (6) u piai xq
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344 Convex Polytopes For x ∈ Bd
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346 Convex Polytopes Rational Polyh
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348 Convex Polytopes Totally Dual I
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350 Convex Polytopes of the vectors
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Geometry of Numbers and Aspects of
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Extension to Crystallographic Group
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Relations Between Different Bases 2
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21 Lattices 359 triangle conv{o,(u,
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By definition of bi+1, Thus (6) yie
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21 Lattices 363 A similar, slightly
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21.4 Polar Lattices 21 Lattices 365
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The First Fundamental Theorem 22 Mi
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22 Minkowski’s First Fundamental
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If 22 Minkowski’s First Fundament
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22 Minkowski’s First Fundamental
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Proof. Consider the linear forms l1
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Proof. It is sufficient to consider
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Thus V (C) ≥ V (O) = 2d � �
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Given C and L, the problem arises t
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23 Successive Minima 383 as require
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24 The Minkowski-Hlawka Theorem 24
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(6) (V (J) ≈) � n�= 0 (7) J
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24 The Minkowski-Hlawka Theorem 389
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The Variance Theorem of Rogers and
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The Selection Theorem of Mahler [68
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26 The Torus Group E d /L 395 where
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Proposition 26.2. E d /L is a compa
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26 The Torus Group E d /L 399 theor
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26 The Torus Group E d /L 401 The m
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The formula to calculate the Jordan
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27 Special Problems in the Geometry
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Asymptotic Estimates 27 Special Pro
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27 Special Problems in the Geometry
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28 Basis Reduction and Polynomial A
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(ii) �b1� ≤2 1 2 (d−1) min
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28 Basis Reduction and Polynomial A
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and so 28 Basis Reduction and Polyn
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Shortest Lattice Vector Problem 28
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and therefore �x − m� 2 = �
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29 Packing of Balls and Positive Qu
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29 Packing of Balls and Positive Qu
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29.3 Error Correcting Codes and Bal
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29 Packing of Balls and Positive Qu
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29 Packing of Balls and Positive Qu
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29 Packing of Balls and Positive Qu
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29 Packing of Balls and Positive Qu
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29 Packing of Balls and Positive Qu
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30 Packing of Convex Bodies 439 The
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30 Packing of Convex Bodies 441 Nex
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Packing and Central Symmetrization
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Existence of Densest Lattice Packin
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30 Packing of Convex Bodies 447 Rem
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A Lower Estimate for the Maximum La
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Lattice Packing and Packing of Tran
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a f b o e 2D c Fig. 30.3. Disc and
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31 Covering with Convex Bodies 455
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31 Covering with Convex Bodies 457
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31 Covering with Convex Bodies 459
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31 Covering with Convex Bodies 461
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32 Tiling with Convex Polytopes 463
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32 Tiling with Convex Polytopes 465
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32 Tiling with Convex Polytopes 467
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32 Tiling with Convex Polytopes 469
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32 Tiling with Convex Polytopes 471
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32 Tiling with Convex Polytopes 473
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Define 32 Tiling with Convex Polyto
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32 Tiling with Convex Polytopes 477
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32 Tiling with Convex Polytopes 479
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33.1 Fejes Tóth’s Inequality for
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Noting that 2h 2 v − hhvv = 2π 2
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33 Optimum Quantization 485 The con
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(7) � (x, y) : x ∈ K, 0 ≤ y
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Next, we prove that (18) lim inf n
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Generalizations 33 Optimum Quantiza
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33 Optimum Quantization 493 set wit
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33 Optimum Quantization 495 In (4)
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33 Optimum Quantization 497 encoder
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34 Koebe’s Representation Theorem
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G 34 Koebe’s Representation Theor
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Di+1 34 Koebe’s Representation Th
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34 Koebe’s Representation Theorem
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34 Koebe’s Representation Theorem
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34 Koebe’s Representation Theorem
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34 Koebe’s Representation Theorem
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References 1. Achatz, H., Kleinschm
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- Page 527 and 528: References 517 88. Beer, G., Topolo
- Page 529 and 530: References 519 136. Bohr, H., Molle
- Page 531 and 532: References 521 190. Carathéodory,
- Page 533 and 534: References 523 237. Dantzig, G.B.,
- Page 535 and 536: References 525 286. Edmonds, A.L.,
- Page 537 and 538: References 527 337. Florian, A., Ex
- Page 539 and 540: References 529 385. Goodman, J.E.,
- Page 541 and 542: References 531 435. Gruber, P.M., C
- Page 543 and 544: References 533 487. Heil, E., Über
- Page 545 and 546: References 535 534. Hyuga, T., An e
- Page 547 and 548: References 537 582. Khovanskiĭ (Ho
- Page 549 and 550: References 539 632. Lazutkin, V.F.,
- Page 551 and 552: References 541 684. Mani-Levitska,
- Page 553 and 554: References 543 736. Minkowski, H.,
- Page 555 and 556: References 545 783. Pach, J., Agarw
- Page 557 and 558: References 547 835. Rickert, N.W.,
- Page 559 and 560: References 549 891. Schmidt, E., Di
- Page 561 and 562: References 551 941. Skubenko, B.F.,
- Page 563 and 564: References 553 992. Teissier, B., V
- Page 565 and 566: References 555 1042. Zassenhaus, H.
- Page 567 and 568: 558 INDEX Bravais classification of
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- Page 571 and 572: 562 INDEX Kneser-Macbeath theorem,
- Page 573 and 574: 564 INDEX rational lattice, 414 rat
- Page 575: 566 INDEX transference theorem of K
- Page 579 and 580: 570 AUTHOR INDEX Giles, J., 1 Giles
- Page 581 and 582: 572 AUTHOR INDEX Kuczma, 13, 17 Kü
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- Page 585 and 586: 576 AUTHOR INDEX Tanemura, 464 Tarj
- Page 587 and 588: 578 List of Symbols sch, schL Schwa
- Page 589: 289. Manin: Gauge Field Theory and