Gruber P. Convex and Discrete Geometry

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20 Linear Optimization 337 particularly strongly related, in the sense that the values of their solutions coincide. Such results are called duality theorems. After some preliminary results on normal cones we present the duality theorem of von Neumann, Gale, Kuhn and Tucker. For more information, see the literature cited earlier. Terminology and Normal Cones Given a linear optimization problem, say sup{cx : Ax ≤ b}, the function x → cx, x ∈ E d is its objective function and the convex polyhedron {x : Ax ≤ b} its feasible set. A point of the feasible set is a feasible solution. Ifthe supremum is attained at a feasible solution, the latter is called an optimum solution. There is an analogous notation for the other linear optimization problems. Let C be a closed convex cone in E d with apex o. Itsnormal cone NC(o) of C at the apex o is the closed convex cone with apex o consisting of all exterior normal vectors of support hyperplanes of C at o, that is, NC(o) = � u : ux ≤ 0 for all x ∈ C � , see Sect. 14.2. Let P be a convex polyhedron. The normal cone NP(p) of P at a point p ∈ bd P is the closed convex cone of all exterior normal vectors of support hyperplanes of P at p. Thenormal cone NP(F) of P at a face F ∈ F(P) is the closed convex cone of all exterior normal vectors of support hyperplanes of P which contain F.Ifp is a relative interior point of F, then it is easy to see that NP(F) = NP(p). An immediate extension of Proposition 14.1 is the following result. Proposition 20.1. Let P = {x : Ax ≤ b} be a convex polyhedron and let p ∈ bd P, resp. F ∈ F(P). Ifa1x ≤ β1,...,akx ≤ βk are the inequalities among the inequalities Ax ≤ b which are satisfied with the equality sign by p, resp. by all x ∈ F, then NP(p), resp. N P(F) = pos{a1,...,ak}. Let F �= ∅be a face of a convex polyhedron P. F is called a minimum face of P if there is no face of P properly contained in F, except the empty face. This means that the polyhedron F has no proper face except for the empty face. Thus the minimum faces are planes of dimension 0 (vertices), 1 (edges, unbounded in both directions), 2,...
338 Convex Polytopes Existence of Solutions Since each convex polyhedron P �= ∅can be represented in the form � λ1 p1 +···+λm pm : λi ≥ 0, λ1 +···+λm = 1 � + � µ1q1 +···+µnqn : µ j ≥ 0 � , the following result is easy to see: Proposition 20.2. Assume that the linear optimization problem sup{cx : Ax ≤ b} has non-empty feasible set P ={x : Ax ≤ b}. Then the supremum is either +∞, or it is finite and attained. An analogous result holds for the problem inf{yb : y ≥ o, yA = c}. The Duality Theorem of von Neumann [768] and Gale, Kuhn and Tucker [353] is as follows. Theorem 20.1. Let A be a real m × d matrix, b ∈ E m , c ∈ E d . If at least one of the extreme sup{cx : Ax ≤ b} and inf{yb : y ≥ o, yA = c} is attained, then so is the other and (1) max{cx : Ax ≤ b} =min{yb : y ≥ o, yA = c}. Proof. We first show the following inequality, where P = {x : Ax ≤ b} and Q ={y : y ≥ o, yA = c}: (2) If P, Q �= ∅, then sup{cx : Ax ≤ b} and inf{yb : y ≥ o, yA = c} both are attained and max{cx : Ax ≤ b} ≤min{yb : y ≥ o, yA = c}. Let x ∈ P, y ∈ Q. Then cx = yAx ≤ yb. Hence sup{cx : Ax ≤ b} ≤inf{yb : y ≥ o, yA ≤ c} and both are attained by Proposition 20.2, concluding the proof of (2). Assume now that P �= ∅and that δ = sup{cx : Ax ≤ b} is attained at p ∈ P = {x : Ax ≤ b}, say. Then p is a boundary point of P and the hyperplane through p with normal vector c supports P at p. Clearly, c is an exterior normal vector of this support hyperplane. Let a1x ≤ β1,...,akx ≤ βk be the inequalities among the m inequalities Ax ≤ b, which are satisfied by p with the equality sign. We may assume that a1,...,ak are the first k row vectors of A and β1,...,βk the first k entries of b. Proposition 20.1 then implies that Thus, (3) c = λ1a1 +···+λkak with suitable λi ≥ 0. and therefore δ = cp = λ1a1 p +···+λkak p = λ1β1 +···+λkβk max{cx : Ax ≤ b} =δ = cp = λ1β1 +···+λkβk = (λ1,...,λk, 0,...,0)b ≥ inf{yb : y ≥ o, yA = c},
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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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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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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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Page 301 and 302: 290 Convex Polytopes P clearly can
Page 303 and 304: 292 Convex Polytopes 17 Rigidity Ri
Page 305 and 306: 294 Convex Polytopes Fig. 17.2. Cau
Page 307 and 308: 296 Convex Polytopes Call the index
Page 309 and 310: 298 Convex Polytopes of V. Fori ∈
Page 311 and 312: 300 Convex Polytopes Thus (3) impli
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Page 315 and 316: 304 Convex Polytopes For other pert
Page 317 and 318: 306 Convex Polytopes where λ is a
Page 319 and 320: 308 Convex Polytopes 18.3 The Isope
Page 321 and 322: 310 Convex Polytopes 19 Lattice Pol
Page 323 and 324: 312 Convex Polytopes Embed E d into
Page 325 and 326: 314 Convex Polytopes To see this, n
Page 327 and 328: 316 Convex Polytopes The Coefficien
Page 329 and 330: 318 Convex Polytopes If the triangl
Page 331 and 332: 320 Convex Polytopes � qP(t) = No
Page 333 and 334: 322 Convex Polytopes Now, take into
Page 335 and 336: 324 Convex Polytopes L � P + (n +
Page 337 and 338: 326 Convex Polytopes V (S) = V .Let
Page 339 and 340: 328 Convex Polytopes (7) ψ|H d = 0
Page 341 and 342: 330 Convex Polytopes Theorem 19.6.
Page 343 and 344: 332 Convex Polytopes (15) L(nP) = L
Page 345 and 346: 334 Convex Polytopes have to be sin
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Page 351 and 352: 340 Convex Polytopes Description of
Page 353 and 354: 342 Convex Polytopes (6) u piai xq
Page 355 and 356: 344 Convex Polytopes For x ∈ Bd
Page 357 and 358: 346 Convex Polytopes Rational Polyh
Page 359 and 360: 348 Convex Polytopes Totally Dual I
Page 361 and 362: 350 Convex Polytopes of the vectors
Page 363 and 364: Geometry of Numbers and Aspects of
Page 365 and 366: Extension to Crystallographic Group
Page 367 and 368: Relations Between Different Bases 2
Page 369 and 370: 21 Lattices 359 triangle conv{o,(u,
Page 371 and 372: By definition of bi+1, Thus (6) yie
Page 373 and 374: 21 Lattices 363 A similar, slightly
Page 375 and 376: 21.4 Polar Lattices 21 Lattices 365
Page 377 and 378: The First Fundamental Theorem 22 Mi
Page 379 and 380: 22 Minkowski’s First Fundamental
Page 381 and 382: If 22 Minkowski’s First Fundament
Page 383 and 384: 22 Minkowski’s First Fundamental
Page 385 and 386: Proof. Consider the linear forms l1
Page 387 and 388: Proof. It is sufficient to consider
Page 389 and 390: Thus V (C) ≥ V (O) = 2d � �
Page 391 and 392: Given C and L, the problem arises t
Page 393 and 394: 23 Successive Minima 383 as require
Page 395 and 396: 24 The Minkowski-Hlawka Theorem 24
Page 397 and 398: (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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29.3 Error Correcting Codes and Bal
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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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32 Tiling with Convex Polytopes 463
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Define 32 Tiling with Convex Polyto
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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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References 1. Achatz, H., Kleinschm
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References 515 36. Arias de Reyna,
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References 517 88. Beer, G., Topolo
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References 519 136. Bohr, H., Molle
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References 521 190. Carathéodory,
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References 523 237. Dantzig, G.B.,
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References 525 286. Edmonds, A.L.,
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References 527 337. Florian, A., Ex
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References 529 385. Goodman, J.E.,
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References 531 435. Gruber, P.M., C
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References 533 487. Heil, E., Über
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References 535 534. Hyuga, T., An e
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References 537 582. Khovanskiĭ (Ho
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References 539 632. Lazutkin, V.F.,
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References 541 684. Mani-Levitska,
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References 543 736. Minkowski, H.,
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References 545 783. Pach, J., Agarw
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References 547 835. Rickert, N.W.,
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References 549 891. Schmidt, E., Di
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References 551 941. Skubenko, B.F.,
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References 553 992. Teissier, B., V
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References 555 1042. Zassenhaus, H.
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558 INDEX Bravais classification of
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560 INDEX existence of elementary v
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562 INDEX Kneser-Macbeath theorem,
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564 INDEX rational lattice, 414 rat
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566 INDEX transference theorem of K
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568 AUTHOR INDEX Björner, 265 Blas
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570 AUTHOR INDEX Giles, J., 1 Giles
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572 AUTHOR INDEX Kuczma, 13, 17 Kü
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574 AUTHOR INDEX Pick, 316 Pisier,
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576 AUTHOR INDEX Tanemura, 464 Tarj
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578 List of Symbols sch, schL Schwa
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289. Manin: Gauge Field Theory and
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