Gruber P. Convex and Discrete Geometry

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Doubly Stochastic and Permutation Matrices 5 The Boundary of a Convex Body 77 A real d × d matrix is doubly stochastic if all its entries are non-negative and the sum of the entries in each row and column equals 1. The set of all doubly stochastic d × d matrices is usually denoted Ωd. It clearly may be considered a subset of Ed2 . Particular cases include the d × d permutation matrices. These are d × d matrices such that, in each row and column, all entries are 0 with one exception which is 1. There are d ! d × d permutation matrices. Birkhoff’s Theorem Because of the importance of doubly stochastic matrices, it is a natural problem to describe the set Ωd of all doubly stochastic d × d matrices as a subset of Ed2 . Birkhoff [119] gave the following satisfying result: Theorem 5.7. The set Ωd of all doubly stochastic d × d matrices is the convex hull of the d × d permutation matrices and each permutation matrix is an extreme point of Ωd. In other words, Ωd is a convex polytope in Ed2 , the vertices of which are precisely the d × d permutation matrices. There exist many proofs of Birkhoff’s theorem. For a proof which makes use of tools from linear programming see, e.g. [841]. A simple geometric proof using properties of convex polytopes, will be given in Sect. 14.2. In the following we present a proof based on the Frobenius–König theorem,wherebya diagonal of a d × d matrix we mean a sequence of d of its elements, one from each row and column. A closely related result is Hall’s marriage theorem. Lemma 5.3. Let M be a real d × d matrix. Then the following are equivalent: (i) Each diagonal of M contains 0. (ii) M has a p × q zero submatrix with p + q = d + 1. Proof. For subsequences σ, τ of the sequence (1,...,d) let Mστ, Mστc, Mσ cτ , and Mσ cτ c denote the matrices obtained from M by cancelling all rows and all columns with indices not in σ and not in τ, not in σ and in τ, inσ and not in τ, and in σ and in τ, respectively. ¬(i)⇒ ¬(ii) Assume that there is a diagonal in M without 0. Let Mστ be a p × q zero submatrix of M. Then, in the columns with indices in τ, the entries of our diagonal must lie in Mσ cτ . Hence q ≤ d − p or p + q ≤ d. (i)⇒(ii) Here the proof is by induction on d. Ford = 1 this implication clearly holds. Assume now that d > 1 and that the implication holds for d − 1. Let M be a real d ×d matrix such that every diagonal contains 0. If M is the zero matrix, there is nothing to prove. Otherwise M has a non-zero entry, say mij.ByassumptiononM, every diagonal of Mic j c contains 0. The induction hypothesis then shows that Mic j c has an r × s zero submatrix with r + s = d. Thus M also has an r × (d − r) zero submatrix. To simplify, permute the rows and columns of M so that the resulting matrix N has the form
78 Convex Bodies N = � � X, O Y, Z where X is an r × r matrix and Z a (d − r) × (d − r) matrix. By assumption on M, every diagonal of N contains 0. Thus, if there is a diagonal of X which does not contain 0, then every diagonal of Z contains 0. This shows that all diagonals of X contain 0 or all diagonals of Z contain 0. We may suppose that the former is the case. (If the latter is the case, the proof is almost identical.) By the induction hypothesis, X has a t × u zero submatrix with t + u = r + 1. Then N, and therefore also M, has a t × (u + d − r) zero submatrix. Since t + u + d − r = r + 1 + d − r = d + 1, the induction is complete. ⊓⊔ Proof of the Theorem. Since a convex combination of doubly stochastic d × d matrices is also doubly stochastic, we see that (1) Ωd is convex. The main step is to prove that (2) Ωd is the convex hull of the d × d permutation matrices. Each doubly stochastic d × d matrix has n positive entries, where d ≤ n ≤ d 2 . For the proof of (2) it is thus sufficient to show the following proposition by induction on n: (3) Let d ≤ n ≤ d 2 . Then each doubly stochastic d × d matrix with n positive entries is a convex combination of d × d permutation matrices. Any doubly stochastic d × d matrix M with precisely d positive entries is a permutation matrix. Thus (3) holds for n = d. Assume now that n > d and (3) holds for n − 1. Let M be a doubly stochastic d × d matrix with n positive entries. We show that M has a diagonal with positive entries. For assume not, that is, each diagonal of M contains 0. Lemma 5.3 then shows that M has a p × q zero submatrix with p + q = d + 1, say Mστ. Since the non-zero entries of M in the rows with indices in σ are in Mστc, the sum of the entries in each row of Mστc is 1. Therefore the sum of all entries of Mστc is p. Similarly, the sum of all entries of Mσ cτ is q. Since Mστc and Mσ cτ are disjoint submatrices of M, the sum of all entries of M is at least p + q = d + 1. This contradiction shows that M has a diagonal with positive entries. Since, by assumption, M has n > d positive entries and the sum of all these n entries is d, among the d entries of the diagonal, there must be one entry less than 1. Let 0
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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
Page 37 and 38: 26 Convex Functions Remark. More pr
Page 39 and 40: 28 Convex Functions even more is tr
Page 41 and 42: 30 Convex Functions G(x). The proof
Page 43 and 44: 32 Convex Functions by (15), unifor
Page 45 and 46: 34 Convex Functions 2.4 A Stone-Wei
Page 47 and 48: 36 Convex Functions x x Fig. 2.1. S
Page 49 and 50: 38 Convex Functions F is convex. Us
Page 51 and 52: 40 Convex Bodies results, character
Page 53 and 54: 42 Convex Bodies Convex Hulls Given
Page 55 and 56: 44 Convex Bodies The next result is
Page 57 and 58: 46 Convex Bodies C ∩ L ⊥ clearl
Page 59 and 60: 48 Convex Bodies Suppose not. Then,
Page 61 and 62: 50 Convex Bodies The Centrepoint of
Page 63 and 64: 52 Convex Bodies Since G is open, w
Page 65 and 66: 54 Convex Bodies in one of the two
Page 67 and 68: 56 Convex Bodies or geometric infor
Page 69 and 70: 58 Convex Bodies v HC (u, h(u)) (v,
Page 71 and 72: 60 Convex Bodies C D Fig. 4.3. Sepa
Page 73 and 74: 62 Convex Bodies For the proof of t
Page 75 and 76: 64 Convex Bodies in contradiction t
Page 77 and 78: 66 Convex Bodies (6) R(t) ⊆ E d i
Page 79 and 80: 68 Convex Bodies 5 The Boundary of
Page 81 and 82: 70 Convex Bodies for all y ∈ C an
Page 83 and 84: 72 Convex Bodies the existence of s
Page 85 and 86: 74 Convex Bodies where r(·) is a s
Page 87: 76 Convex Bodies Hence x ∈ conv e
Page 91 and 92: 80 Convex Bodies Further topics of
Page 93 and 94: 82 Convex Bodies We are now ready t
Page 95 and 96: 84 Convex Bodies similarly, D j is
Page 97 and 98: 86 Convex Bodies For the proof of t
Page 99 and 100: 88 Convex Bodies The definition of
Page 101 and 102: 90 Convex Bodies If ei ∈ Pi is no
Page 103 and 104: 92 Convex Bodies Let v(·) denote (
Page 105 and 106: 94 Convex Bodies The present sectio
Page 107 and 108: 96 Convex Bodies Proposition 6.5. L
Page 109 and 110: 98 Convex Bodies Thus Second, Minko
Page 111 and 112: 100 Convex Bodies The Valuation Pro
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Page 115 and 116: 104 Convex Bodies To remedy the sit
Page 117 and 118: 106 Convex Bodies Proof. Applying S
Page 119 and 120: 108 Convex Bodies Third, let C be a
Page 121 and 122: 110 Convex Bodies A Problem of Sant
Page 123 and 124: 112 Convex Bodies sets of S. Ifφ c
Page 125 and 126: 114 Convex Bodies = � (−1) |J|
Page 127 and 128: 116 Convex Bodies If P has dimensio
Page 129 and 130: 118 Convex Bodies Since, by Volland
Page 131 and 132: 120 Convex Bodies that is, V is sim
Page 133 and 134: 122 Convex Bodies Let C, D ∈ C su
Page 135 and 136: 124 Convex Bodies P Q . . . . Fig.
Page 137 and 138: 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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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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29.3 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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