notation index - cambridge university...

Notation index

+∞, 490+ϕ, 496+p, 490−A, xxA + B, xxA − B, xxA | E, xxiiA ÷ B, 146A ⊕ B, xxA ∨ B, 33A•, 329C∗, 35K◦, 32K�, 507Kρ, 209[L, L′], 240[·, ·], xix#, 459•⋃, 336‖ · ‖K , 53≈S , 347≈T , 347f̄ , xxii·p, 490η � η′, 251〈 f 〉, 611〈·, ·〉, xixν ∗ μ, 460⊕M , 495IIu, 116⊂, xx+̂p, 512

#̂, 513F̂, 75+̃, 507#̃, 509{ f < α}, 19{ f = α}, 19

{ f ≤ α}, 19f ∗ μ, 177f � g, 517f ∗, 37f •, 41f ◦, 5231A, xxIIx, 114Ix, 114

A(n, k), xxiiAρ(K, β), 121A, 42a(K, ·), 59aff , xix

B(K), 421B(z, ρ), xxiBn, xxiB0(z, ρ), xxiB(X), 209B(K), 590bi j, 116bd, xx

C+(Sn−1), 552C(Sn−1), 177C∞, 113Ck, 113Ck+, 115Cα(Rn), 522Cm(K, ·), 214Cvx(Rn), 40Cvx0(Rn), 41CK, 547C(X), 69Cn, 60CCn, 42CCno, 33C n, 35c(K), 314c(K, β, u), 234

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Cambridge University Press978-1-107-60101-7 - Encyclopedia of Mathematics and its Applications: Convex Bodies: The Brunn–Minkowski Theory: Second Expanded EditionRolf SchneiderIndexMore information

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http://www.cambridge.org/9781107601017http://www.cambridge.orghttp://www.cambridge.org

716 Notation index

cρ(K, η, x), 231cl, xxconv, 2χ, 214, 230

D(K), xxi, 49D(·, . . . , ·), 124, 322Dk(x1, . . . , xk), 303DK, 140, 529Du f , 610d(A, x), xxid(K, K′), 250d(K, ·), 26dBM, 589diam, xxidim, 7dom, 20δ, 61δQ, 71δS , 71δD, 71δp, 71δw, 71δ̃, 507Δ, 119Δ(K), 49ΔS , 119∂, 29

EpK, 592, 593E(K), 75EL(K), 587EJ(K), 587Ei(K), 75Ep( f ), 610ei j, 116epi, 20exp, 18extr, 17ext, 17expr , 76extr , 76ηr , 79

F(K, ·), 45F (K), 74F n, 545F nc , 546F n(o), 545Fi(K), 74f (K, ·), 545Φ̃i(K), 515Φi(K), 515Φ

r,sk , 317

Φm(K, ·), 216G(K), 549G(n, k), xxii

Gδ, 132Gp(K), 555Gn, xxiiGL(n), xxiig(K, ·), 53gK , 528ΓφK, 568ΓpK, 567Γ−pK, 575γ(F, P), 109γn, 602γ(F, F′, P, P′), 240

H(K, ·), 45H−(K, ·), 45H+u,α, xxH−u,α, xxH j, 115Hu,α, xxHess, 28Hk, xxih(K, ·), 44h∗(K, ·), 262I∞A , xx, 21I+p , 582Ip, 581IK, 580i(K, q, u), 234int, xx

J , 42j(K, q, x), 231

Ks, 546Ku, 302K f , 51K[δ], 560Kns,1, 164Kns , 164K(A), 8Kn, 8Kn(r,R), 91Knc , 546Knn , 8Kn(o), 32Kn(os), 463Knos, 495Kno , 32KC , 440Kn(A), 8Knos, 290ki(x, u), 129κn, xxiκi(x, t), 113κs(x, t), 113

L(F), 240LK , 605



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Notation index 717

L2(Sn−1), 623LC(Rn), 517L, 40L(K), 590lin , xix (K), 602ΛK, 546Λm(K, ·), 216λF , 240

Mk(K), 265M(Sn−1), 177M(Sn−1, p), 177MK, 566MVal, 364MValm, 364M n, 615m(L ⊂ K), 418μ, xxii, 237μρ(K, ·), 209μk , xxii, 237

N(K, F), 83N(K, x), 81N(P), 348Nor K, 127, 209Nor , 59ν, xxii, 237νk , xxii, 237∇, 26, 47o, xixΩ(K), 544, 545Ωφ, 556Ωp(K1, . . . , Kn), 559

Ω̃−p(K), 559Ωp(K), 554ωn, xxi

P(K, ·), 81projE , xxiiPK, 581P(A), 348Pn, 8, 104Pnn, 8, 104p̃, 325p(A, x), 9pr(K), 314pos, 2ΠK, 302, 569Π〈 f 〉, 611Π◦K, 570ΠφK, 575ΠpK, 575Ψr , 316Ψm(K, ·), 216Q, 316

Qn, 523qr(K), 314

R(A, x), 9RpK, 574Rn, xixR̄, 19rec, 16regn, 87relbd, xxrelint, xxρ(A), 143ρi(x, t), 113ρs(x, t), 113

S (K, ·), 81S (·, . . . , ·, ·), 279, 280S m(K, ·), 214SH K, 536SK , 465SO(n), xxiiSL(n), xxiiSV , 348S(K), 163Snc , 546Sno, 57, 507Sn−1, xxiS m, 623s(K), 50s j, 117s(K), 546Σ, 209σ, xxi, 469σ(K, β), 88

T (K, M), 262T (K, u), 85T (u1, . . . , un−1), 304TxK, 112Θm(K, η), 212τ, xixτ(K, ω), 88

U(K), 91U(S ), 330U•(S ), 332Ui j, 401U(S ), 332u(A, x), 9uK , 83, 88, 113

V(·, . . . , ·), 277, 280V1(K, L), 286VK (ω), 501V j(K), 208, 214Vp(K, L), 492V(i), 406V(i)(K, L), 381Vi j, 421



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718 Notation index

VK , 495Ṽ , 508Ṽi, 508Val∞, 366Val∞m , 366Val, 353Val+m, 353Val−m, 353ValG , 367Vn, 547Vnc , 547�(k), 303vert, 105vr(K), 597

Wi(K), 208, 213Wi(·, ·), 427Wx, 113

W̆i, 514Wu, 116Ŵi(K), 514W̃n−i, 508w(K), 50, 297w(K, ·), 49xK , 88Ξ

(k)m (K, ·), 227

ξK , 115

Υ, 48

Zφ(Λ), 609Z+p (Λ), 609Z ns , 341z(·, . . . , ·), 313zr+1(K), 312



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Author index

Abardia, J., 367, 580, 622Aczél, J., 349Ader, O. B., 589Adiprasito, K., 137Ahrens, I., 152Alberti, G., 89Aleksandrov, A. D., xi, xvi, 31, 58, 60, 89, 90,

112, 127, 131, 202, 226, 274, 286, 287, 295,309, 327, 385, 398, 408, 409, 417, 438, 439,454, 461, 464, 472, 477, 486–488, 498, 505,628

Alesker, S., 60, 201, 206, 295, 361, 363, 365–368,398, 578

Alexander, R., 205Alexandrov, V., 462Alfsen, E. M., 18Allendoerfer, C. B., 223Alonso-Gutiérrez, D., 381, 578, 606, 609,

614Alvino, A., 390Ambartzumian, R. V., 204Ambrosio, L., 89Ambrus, G., 541Anderson, R. D., 89Andrews, B., 504, 506, 558Anikonov, Yu. E., 628Arnold, R., 71, 320, 428, 439, 487Arocha, J. J., 8Arrow, K. J., 150, 151Artstein, Z., 150, 152, 154Artstein-Avidan, S., 40, 42, 43, 182, 518, 524,

525Ash, R. B., 209Asplund, E., 32, 76Assouad, P., 205Atkinson, K., 623Auneau-Cognacq, J., 265, 321Averkov, G., 529Avriel, M., 520

Baddeley, A. J., 236Baebler, T., 390Bair, J., 15, 19, 153, 154, 171Balashov, M. V., 167Ball, K., 374, 380, 523, 532, 573, 577, 588, 589,

597, 599, 600, 602, 604Bambah, R. P., 565Banchoff, T. F., 235Bandle, C., 391Bandt, Ch., 72, 73Bangert, V., 31, 274Bantegnie, R., 72Bapat, R. B., 328Baraki, G., 73Bárány, I., 8, 104, 112, 137, 462, 558, 562Barthe, F., 377, 381, 525, 565, 598–600, 602Barthel, W., 390, 391, 580Barvinok, A., xv, 594Bastero, J., 380, 594, 606, 614Batson, R. G., 19Bauer, C., 170, 439, 454, 472Bauer, H., 18, 209Baum, D., 59Bayen, T., 201Bazylevych, L. E., 70Beer, G. A., 69Behrend, F., 589, 593, 596, 600, 603Bensimon, D., 78Benson, R. V., xv, 393Benyamini, Y., 391Berck, G., 581Berg, Ch., 181, 186, 470, 628Berger, M., xvBernig, A., 228, 236, 261, 366–368, 578, 580, 622Bernštein, D. N., 398Bernstein, J., 361, 367Bernués, J., 380, 606, 614Besicovitch, A. S., 89, 103Betke, U., 202, 204, 311, 386Bettinger, W., 390



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720 Author index

Bianchi, G., 31, 529, 532, 540, 541Bianchini, C., 532, 543Biehl, Th., 540Bisztriczky, T., xvii, 541, 569Björck, G., 77Blackwell, D., 205Blagojević, P. V. M., 8Blaschke, W., xii, xv, 69, 70, 126, 164, 181,

201–204, 261, 308, 320, 361, 378, 388, 463,472, 473, 538, 540, 541, 543, 545, 546, 548,550, 560, 591, 595, 628

Blind, R., 487Blumenthal, L. M., 72Bobkov, S. G., 520, 522, 523, 525, 614Böhm, J., 338Böröczky, K., 462Böröczky, K. J., 34, 380, 389, 393, 445, 495, 504,

506, 525, 534, 541, 558, 565, 566, 569, 576,605

Bokowski, J., 264, 310, 389Bol, G., 149, 151, 388, 389, 417, 432, 444, 628Bolker, E. D., 201–205Boltyanski, V., xvi, 201, 418Boman, J., 126Bonnesen, T., xi, xv, xvi, 58, 60, 81, 90, 91, 155,

188, 286, 287, 308, 320, 370, 378, 386, 388,393, 461

Bonnice, W., 15Borell, C., 380, 517, 520Borowska, D., 167Borwein, J. M., xvi, 31, 144, 151Bose, R. C., 319, 320Botts, T., 9Bourbaki, N., 15, 18, 79, 90Bourgain, J., 181, 203, 204, 378, 387, 486, 540,

541, 565, 585, 605, 628Boček, L., 393Bracho, J., 8Brandolini, L., xviiBrannen, N. S., 389, 578Brascamp, H. J., 380Brehm, U., 236Brickell, F., 545Bröcker, L., 236, 367Brøndsted, A., xvi, 19, 90, 112Bronshtein, E. M., 72, 73, 79, 172, 463Brooks, J. N., 168Brown, A. L., 78Brunn, H., xi, 377, 535Buchta, C., 562Budach, L., 236Bunt, L. N. H., 10Burago, Ju. D., 10, 15, 274, 379, 390, 398Burchard, A., 540Burckhardt, J. J., 202

Burton, G. R., 80, 168, 202Busemann, H., xvi, 31, 125, 127, 131, 274, 327,

390, 391, 398, 454, 462, 487, 515, 536, 539,580, 583

Caffarelli, L. A., 380, 462Campi, S., 379, 463, 473, 540, 541, 543, 566, 568,

569, 608, 628Cannarsa, P., 89Carathéodory, C., 3, 8, 540Carlen, E. A., 605Cascos, I., 206Cassels, J. W. S., 150Chai, Y. D., 516Chakerian, G. D., 201, 203, 393, 417, 454, 463,

477, 530, 532, 534, 586Chang, S.–Y. A., 399Chavel, I., 390, 391Cheeger, J., 236Chen, B., 295, 339, 361Chen, F., 569Chen, L.–Y., 393Chen, W., 506, 558Cheng, S.–Y., 462Chern, S. S., 125, 261, 309, 454Chernoff, P. R., 628Cheung, W.–S., 393, 418, 512, 559, 578, 579, 583,

606Choquet, G., 18, 79, 202, 205Chou, K.–S., 462, 500, 506Cianchi, A., 613Cieślak, W., 628Clack, R., 606Coifman, R. R., 623Colesanti, A., 31, 89, 228, 273, 379, 380, 463,

519, 520, 522, 523, 525, 532, 540, 541, 543Collier, J. B., 79Colzani, L., xviiCordero-Erausquin, D., 605Corson, H. H., 79Coupier, D., 541Coxeter, H. S. M., 202Cressie, N., 154Cuoghi, P., 380Curtis, D. W., 70Czipszer, J., 417

D’Agata, A., 151Dalla, L., 79, 80, 541Danzer, L., 8, 15, 587Dar, S., 379, 398Das Gupta, S., 380Davis, C., 15Davy, P. J., 321Davydov, Yu., 541de Rham, G., 137Debrunner, H., 310



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Author index 721

Debs, G., 78Deicke, A., 545, 546Delgado, J. A., 168Dierolf, P., 69Dinghas, A., 71, 152, 310, 378, 379, 388, 390, 417

418, 444, 628Diskant, V. I., 378, 384, 387, 389, 391, 409, 417,

445, 478, 482, 484, 486, 487Dolzmann, G., 552Dor, L. E., 205Dou, J., 506Drešević, M., 69Duan, X., 516Dubins, L. E., 19Dubois, C., 154Dubuc, S., 380Dudley, R., 32, 72Dunkl, C. F., 179Duporcq, E., 320Dvoretzky, A., 565Dyer, M., 289, 312Dziechcińska-Halamoda, Z., 155

Eckhoff, J., 8, 339Edelstein, M., 79, 154Efimow, N. W., 439Eggleston, H. G., xv, 79, 80, 388, 532, 540Egorychev, G. P., 327Eifler, L. Q., 78Ekeland, I., 150Epelman, M. S., 90Evans, L. C., 31Ewald, G., 60, 91, 103, 104, 152, 170, 429

Fabila, R., 8Falconer, K. J., 628Faro Rivas, I., 188Fáry, I., 287, 417, 462, 543, 561Favard, J., 89, 90, 389, 393, 409, 410, 431, 438,

439, 444, 472Federer, H., 31, 89, 224, 226, 236, 261, 262, 272,

273, 390Fedotov, V. P., 79, 90, 310, 477, 497Fejes Tóth, L., 389, 392, 541, 558Feller, W., 31, 127, 131Fenchel, W., xi, xv, xvi, 58, 60, 81, 90, 91, 155,

188, 223, 286, 287, 308, 320, 370, 378, 386,398, 409, 439, 454, 461

Feng, Y., 579Ferone, V., 390Ferrari, F., 410Ferrers, N. M., 320Fesmire, S., 154Figalli, A., 377, 383, 392Figiel, T., 203, 391Fillastre, F., 527Filliman, P., 202, 439

Fillmore, J. R., 473, 628Firey, W. J., 59, 60, 80, 126, 167, 168, 188, 203,

263, 273, 288, 310, 387, 393, 463, 470, 473,477, 490, 497, 504, 510, 512, 513

Fish, A., 583Fisher, J. C., 628Flaherty, F. J., 236Flanders, H., 319, 389Flatto, L., 101Fleury, B., 569Florentin, D., 43Florian, A., 71, 392Focke, J., 201, 628Fogel, E., 154Folkman, J., 141, 150Fourneau, R., 15, 154, 169, 533Fradelizi, M., 409, 525, 543, 564, 565Fragalà, I., 519, 520, 522, 523Franchi, B., 410Franz, G., 580Freiman, G. A., 445Fresen, D., 563Fu, J. H. G., 31, 228, 236, 261, 262, 366–368Fuglede, B., 19, 388, 391, 628Fujiwara, M., 90, 201, 628Fukuda, K., 154Fusco, N., 391, 392, 628

Gage, M. E., 388, 504, 506Gale, D., 59, 167, 168Gallego, E., 367, 410, 527Gallivan, S., 79Gao, F., 228, 410Gårding, L., 323, 327Gardner, R. J., xvi, 79, 152, 290, 369, 374, 377,

379, 411, 418, 454, 463, 491, 495–497, 510,511, 516, 532, 535, 539, 572, 574, 576, 582,584, 586, 610

Gariepy, R. F., 31Gates, J., 364Geivaerts, M., 152, 167, 168Gensel, B., 201Geppert, H., 444, 526, 628Gericke, H., 319, 410, 628Ghandehari, M., 393Giannopoulos, A., 409, 541, 565, 569, 594, 603,

604, 606Giné, E., 73, 154Glasauer, S., 104, 225, 227, 262Gluck, H., 462Godbersen, C., 532, 533Godet-Thobie, Ch., 70Görtler, H., 289, 628Goikhman, D. M., 463Goldberg, M., 201



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722 Author index

Goodey, P. R., 168, 201, 202, 204, 206, 207, 227,262, 265, 273, 287, 289, 308, 311, 438, 463,473, 477, 478, 529, 582, 584, 628

Goodman, A. W., 320Goodman, J. E., xviiGordon, Y., 203, 525, 565, 566, 594Góźdź, S., 628Gray, A., 223Green, J. W., 60, 603Green, M., 225Grinberg, E. L., 188, 471, 515, 582, 586Gritzmann, P., 154, 202, 289, 312, 408, 409,

462Groemer, H., xvi, 132, 201, 264, 288, 292, 293,

332, 334, 338–340, 376–378, 383, 386, 387,391, 392, 421, 438, 539, 623, 628

Gromov, M., 52, 60, 378, 398, 612Gronchi, P., 379, 463, 473, 540, 541, 543, 566,

568, 569, 608Gross, W., 538, 540Gruber, P. M., xv, xvi, 31, 42, 71–73, 112,

136–138, 167, 411, 418, 533, 536, 540, 558,588, 594

Grünbaum, B., xvi, 8, 15, 73, 79, 80, 112, 155,169, 181, 186, 463, 477, 532

Grynkiewicz, D., 445Grzaślewicz, R., 171Grzybowski, J., 151, 167, 168, 170, 171, 472Guédon, O., 569Guan, B., 478Guan, P., 399, 478Guggenheimer, H., xvi, 487, 564Guilfoyle, B., 201Guleryuz, O. G., 595Guo, Q., 155Gurvits, L., 327Gustin, W., 600

Haberl, C., 504, 557, 568, 575, 582, 585, 613,620, 621

Hadamard, J., 125Hadwiger, H., xii, xv, 69, 151, 173, 181, 182, 202,

223, 235, 239, 261, 264, 310, 319, 320, 338,339, 341, 346–348, 360–363, 378, 379, 380,387–390, 392, 393, 398, 417, 440, 444, 514,540

Hahn, F. H., 150, 151Hahn, M. G., 73, 154Halmos, P. R., 205Halperin, D., 154Hammer, P. C., 153, 188Han, W., 623Hann, K., 125, 225Haralick, R. M., 151Hartenstine, D., 512Hartman, P., 126

Hartzoulaki, M., 409, 541Hausdorff, F., 68, 69Hayashi, T., 320, 628He, B., 506, 559, 576, 583, 587, 606Heil, E., 69, 201, 298, 310, 389, 390, 393, 565Heine, R., 399Helly, E., 4, 8Henk, M., 224, 506, 588Herglotz, G., 223, 463Hernández Cifre, M. A., 224, 225, 393, 440Hertel, E., 338Heveling, M., 224Hilbert, D., 378, 398Hildenbrand, W., 69, 151, 205, 207Hille, E., 58Hinderer, W., 227, 361Hiriart-Urruty, J.–B., 32Hirose, T., 69Höbinger, J., 536Hörmander, L., xvi, 42, 70, 323, 327, 398Hoffman, L. M., 73Holický, P., 79Holmes, R. B., 15Holmes, R. D., 391Horn, B. K. P., 462Howard, R., 202, 454, 558Howe, R., 137Hu, C., 506Hu, Y., 497, 566Huang, Q., 505, 583, 609Huang, Y., 506Huck, H., 309Hufnagel, A., 289, 312, 462Hug, D., 89, 125, 127, 132, 152, 205, 224, 225,

227, 228, 262, 273, 274, 290, 312, 321, 361,363, 364, 379, 410, 454, 463, 482, 486, 491,495–497, 500, 510, 532, 552, 555, 564, 565,594, 604

Hurwitz, A., 472, 628Husain, T., 154

Inzinger, R., 628Iosevich, A., xviiIvaki, M. N., 506Ivanov, B. A., 104Ivanov, G. E., 168

Jacobs, K., 18Jaglom, I. M., xvi, 201Jenkinson, J., 595Jensen, E. B. V., 265, 321, 511Jerison, D., 380Jessen, B., 31, 60, 125, 223, 286, 287, 409, 454,

461Jetter, M., 225Jiang, J., 497Jiang, M. Y., 506



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Author index 723

Jiménez, C. H., 381, 589Jin, H. L., 155John, F., 588, 591Jonasson, J., 534Jongmans, F., 18, 70, 72, 89, 154, 171Jourlin, M., 155Juhnke, F., 588

Kaasalainen, M., 462Kahn, J., 399Kakeya, S., 201, 628Kalinin, B. V., 274Kallay, M., 60, 112, 169, 171, 472Kalman, J. A., 19Kaluza, T., 151Kameneckii, I. M., 201, 628Kampf, J., 223Karasëv, R. N., 8, 167Karavelas, M. I., 154Karcher, H., 168Katsnelson, V., 224, 225Katsurada, Y., 309Kelly, P. J., xvKendall, W. S., 151Kenderov, P., 137Khovanskiı̆, A. G., 327, 340, 363, 398Kiderlen, M., 179, 182, 460, 463, 532Kim, J., 566, 583Kincses, J., 167Kiselman, C. O., 126Klain, D. A., 340, 355, 356, 360, 362, 364, 378

462, 511, 540Klartag, B., 182, 328, 399, 519, 525, 540, 569,

594, 605Klee, V., xvi, 8, 15, 18, 19, 59, 78–80, 89, 103,

136, 153, 155, 167, 272, 289, 339, 532Klein, E., 69Klima, V., 137Klingenberg, W., 201Klötzler, R., 201Kneser, H., 59, 370, 378, 463Kneser, M., 224Knothe, H., 372, 378, 388, 393, 439, 628Knyazeva, M., 526König, H., 586Kohlmann, P., 236, 310, 487Koldobsky, A., xvi, 576, 582–584, 586Kone, H., 622Kopteva, N., 462Koutroufiotis, D., 168, 487Krantz, S. G., 126Krein, M., 18Kropp, R., 227Kruskal, J. B., 78Kubota, T., 319, 320, 409, 438, 472, 628Kühnel, W., 236

Kuiper, N. H., 236Kuperberg, G., 565, 578Kuppe, M., 236Kuratowski, C., 69Kutateladze, S. S., 462, 463

Lachand-Robert, T., 201Laczkovich, M., 79Lafontaine, J., 236Lagarias, J. C., 327Laget, B., 155Lamberg, L., 462Langevin, R., 526Larman, D., 8, 78–80, 90, 91, 103, 104, 477, 541,

562Lashof, R. K., 125Last, G., 224Laugwitz, D., 57, 587Lay, S. R., xvLebesgue, H., 201Ledoux,M., 614Lee, Y. S., 516Lehec, J., 525Leichtweiß, K., xv, xvi, 59, 287, 327, 398, 438,

454, 487, 526, 543, 548, 550, 553, 560, 561,564, 589, 595

Leindler, L., 373Leng, G., 380, 454, 506, 511, 513, 514, 516, 559,

565, 569, 576, 579, 582, 583, 585, 586, 593,595, 606, 609

Lenz, H., 339, 587Letac, G., 472Lettl, G., 71, 72Levin, B. Ja., 472Levitt, G., 526Lévy, P., 205Lewis, D. R., 588, 594, 607Lewis, J. E., 152Lewy, H., 462Li, A.–J., 569, 603, 606Li, A.–M., 543Li, D.–Y., 418Li, J., 399, 478Li, K., 506, 576Li, N., 559Li, X., 511Li, Y., 478, 506Liapounoff, A. A., 205Lieb, E. H., 380Lin, C., 478Lin, Y., 565Lindelöf, L., 385Lindenstrauss, J., 181, 203–205, 378, 387, 391,

486, 540, 591, 628Lindquist, N. F., 202, 206Linhart, J., 202, 204



ExcerptIndex


724 Author index

Linke, Ju. E., 73Little, J. J., 462Litvak, A. E., 534, 594Liu, L., 587Ljašenko, N. N., 154Lonke, Y., 203, 582Lopez, M. A., 566Loritz, A., 439Lü, S.–J., 586Lu, F., 506, 513, 516, 559, 583, 595Lu, G., 410Lu, J., 506Lu, Q., 506Ludwig, M., 532, 554, 556–558, 568, 575, 582,

613–616, 618, 620–622Lutwak, E., xvii, 288, 389, 393, 454, 460, 463,

490, 495–498, 500, 504–507, 510, 512–516,540, 546, 549–551, 554, 558, 559, 564, 567,568, 570–572, 575–578, 580–582, 584–586,591, 592, 595, 600, 601, 603, 605, 607,610–613

Lv, S., 454, 511, 559, 579, 586, 595Lyusternik, L. A., xvi, 390

Ma, T.–Y., 586, 595Ma, X.–N., 399, 478, 506Macbeath, A. M., 73, 539Maehara, H., 160Maggi, F., 377, 383, 392Mahler, K., 564, 565Makai, E., 223Makai, E., Jr, 462, 533, 541, 564, 566Malkevitch, J., xviiMani-Levitska, P., 15, 79, 261, 331, 332, 339, 541Mao, W., 583Marchaud, A., 535Maresch, G., 609Markessinis, E., 606Marti, J. T., xvi, 10, 15, 31, 89Martin, K., 78Martinez-Maure, Y., 488, 526Martini, H., 73, 201, 202, 418, 532, 541, 578, 582Mase, S., 154Matheron, G., 69, 151, 168, 202, 205, 225, 235,

236, 289, 308, 417, 529Matoušek, J., 8Matschke, B., 8Matsumura, S., 409, 628Matveev, V. S., 595Mazur, S., 89McAllister, B. L., 69McCann, R. J., 32, 378McClure, D. E., 558McKinney, R. L., 15McMinn, T. J., 103

McMullen, P., xi, xvi, xvii, 9, 59, 72, 73, 76, 112,131, 155, 165, 167–169, 171, 202, 204, 288,289, 319, 321, 332, 339, 340, 346, 352, 353,362, 366, 378, 461, 463, 533

Mecke, J., 151Meier, Ch., 295, 319, 346Meissner, E., 201, 319, 320, 628Melzak, Z. A., 137Menger, K., 72Meschkowski, H., 152Meyer, M., 381, 409, 438, 525, 530, 536, 541,

543, 556, 560, 562–566, 583, 594Meyer, P., 223Meyer, W., 112, 167, 169, 181Michael, E., 69Milka, A. D., 171Milman, D., 18Milman, E., 569Milman, V., 33, 40–43, 181, 182, 203, 288, 380,

391, 398, 409, 518–520, 523–525, 540, 541,565, 580, 591, 594, 603, 604, 606, 628

Minkowski, H., xi, 18, 90, 151, 155, 188, 286,287, 320, 377, 385, 386, 431, 432, 439, 453,455, 461, 463, 628

Minoda, T., 628Miranda, C., 462Molchanov, I., 150, 154, 206Montejano, L., 8, 70, 167Moore, J. D., 487Moreno, J. P., 167Mosler, K., 206Moszyńska, M., xv, 507Motzkin, T. S., 10Müller, C., 623Müller, H. R., 320, 321Müller, W., 236Mürner, P., 202

Nádenı̂k, Z., 393, 477Nadler, S., 70Nakajima, S., 320, 628Naszódi, M., 589Nazarov, F., 204, 565, 583Netuka, I., 137Neveu, J., 240Newman, D. J., 101Neyman, A., 207Nikliborc, W., 320Nirenberg, L., 462Nitsch, C., 390Noll, D., 31, 132

O’Brien, R. C., 144, 151Oda, T., 224Ohmann, D., 223, 379, 387, 388, 390, 410Oishi, K., 628Oliker, V. I., 454, 464, 486, 506



ExcerptIndex


Author index 725

Oliveros, D., 8Osher, S., 225Oshio, S., 417Osserman, R., 388–391Oudet, É., 201Overhagen, T., 310

Pach, J., 462Pajor, A., 541, 564, 580, 591, 594, 604, 606Pallaschke, D., 170, 288Palmon, O., 589Panina, G. Yu., 204, 293, 340, 488, 526Panov, A. A., 328Paouris, G., 310, 409, 541, 565, 569, 582, 583,

595, 606Papaderou–Vogiatzaki, I., 263Papadimitrakis, M., 603Papadopoulou, S., 78Parapatits, L., 361, 365, 463, 557, 579, 618, 619,

621Parks, H. R., 126Pastor, J. A., 393Pavlica, D., 103Payá, R., 169Pelczyński, 588Peña, A., 380Perissinaki, I., 594Perles, M. A., 235, 338Petermann, E., 393Petrov, F., 565Petty, C. M., 202, 203, 536, 546–549, 564, 567,

569, 570, 572, 579, 581, 583, 591, 594, 603,628

Pfiefer, R. E., 539Phelps, R. R., 18, 186Phillips, R. S., 58Pisier, G., 374, 380, 565Pivovarov, P., 310, 569Pogorelov, A. V., xvi, 274, 461, 462, 471–473,

486, 487Pollack, R., xviiPolovinkin, E. S., 160, 167Pólya, G., 391Pompeiu, D., 68Ponsiglione, M., 392Pontryagin, L. S., 151, 181Positsel’skii, E. D., 174, 181Pranger, W., 19Pratelli, A., 377, 383, 392, 628Prodromou, M., 558Protasov, V. Ju., 495Przesławski, K., 52, 60, 73, 287, 322, 339Przybycień, H., 151, 167Pucci, C., 31, 89Pukhlikov, A. V., 340, 363

Qi, C., 559

Quinn, J., 70

Rédei, L., 561Rényi, A., 558Rademacher, H., 58Radon, J., 3, 8, 188Rådström, H., 70Rajala, K., 392Rataj, J., 227, 236, 265, 312Ratschek, H., 70Rauch, J., 126, 168Reay, J. R., 15Rédei, L., 543Reidemeister, K., 89, 204Reisner, S., 381, 438, 525, 530, 536, 543, 560,

564–566, 583Reiter, H. B., 78Reitzner, M., 104, 112, 554, 556–558, 562Ren, D., 534Renegar, J., 323, 327Rényi, A., 417Rešetnjak, Ju. G., 31, 487, 628Reuleaux, F., 201Ricker, W., 207Rickert, N. W., 202, 205Riesz, F., 59Roberts, A. W., xi, xvi, 31Rockafellar, R. T., xi, xv, 9, 19, 30–32, 41,

42, 59Rodrı́guez, L., 526Rogers, C. A., xii, 69, 79, 91, 103, 104, 169, 477,

530, 532, 533, 541, 565Romance, M., 594, 606Rosenberg, H., 526Rota, G.–C., 339, 364Rotem, L., 43, 519, 520, 522–524Rother, W., 261, 264Roy, A. K., 154Roy, N. M., 18Roy, S. N., 319, 320Rubin, B., 584Rubinov, A. M., 463Rudelson, M., 603Ryabogin, D., 204, 565, 580, 583, 585

Sacksteder, R., 125Sah, C.–H., 360Saint Pierre, J., 73, 181, 188Saint Raymond, J., 564Saks, M., 399Salani, P., 380Salinas Martı́nez, G., 393Salinetti, G., 69Salkowski, E., 543Sallee, G. T., 131, 151, 160, 166, 167, 169, 181,

235, 332, 338–340Sandgren, L., 59



ExcerptIndex


726 Author index

Sangwine-Yager, J. R., 223–226, 387, 389, 417,439

Santaló, L. A., 223, 261, 310, 388, 393, 410, 546,548

Saorı́n, E., 224, 225, 379, 440Saroglou, C., 541, 543, 578, 606Sas, E., 541, 628Saškin, Ju. A., 18Schaal, H., 201Schechtman, G., 391, 602Schirokow, A. P., 543Schirokow, P. A., 543Schmidt, E., 378, 391Schmidt, K. D., 70Schmitt, K.–A., 152Schmuckenschläger, M., 381, 530, 561, 577, 602Schneider, R., xv, xvii, 34, 35, 60, 71–73, 78, 80,

90, 104, 112, 126, 137, 144, 151, 155, 167,168, 170, 171, 179–182, 188, 190, 198,200–203, 205, 206, 224, 226, 235, 236,261–264, 273, 274, 287, 288, 295, 309, 312,320, 321, 332, 338, 358, 362–364, 391, 399,409, 410, 417, 421, 428, 438, 439, 444, 450,454, 461, 463, 471–473, 477, 482, 484,486–488, 497, 516, 529, 533, 535, 540, 545,558, 578, 582, 583, 594, 604, 628

Scholtes, S., 170, 288, 472Schori, R. M., 70Schröder, G., 70Schrader, R., 236Schröcker, H.–P., 594Schürger, K., 154Schürmann, A., 506Schütt, C., 525, 552, 556, 558, 561–563, 566Schuster, F. E., 179, 182, 200, 201, 206, 364–366,

368, 461, 463, 512, 568, 575, 578, 579, 586,588, 594, 601, 609, 613, 619, 628

Schuster, R., 321, 363Schwarz, T., 137Schwella, A., 409Seeger, A., 32Seeley, R. T., 119, 623Segal, A., 42, 43, 377, 379, 384Segura Gomis, S., 393Sen’kin, E. P., 473Serra, J., 151Serra, O., 445Shahin, J. K., 310Shapley, L., 141, 150Shen, C., 506Shen, Y., 583Sheng, W., 478, 558Shephard, G. C., xi, xvi, 9, 15, 71, 72, 112, 152,

164, 167–170, 181, 186, 202, 235, 319, 340,399, 530, 532, 533, 541, 583, 628

Si, L., 580, 595Silverman, R., 169Simon, U., 309, 310, 543Slomka, B. A., 42, 43Smilansky, Z., 169Soberón, P., 8Solanes, G., 228, 367, 410, 527Soltan, P. S., 418Soltan, V., 532Song, X., 535Sorger, H., 137Sorokin, V. A., 152Spiegel, W., 69, 332, 346Stachó, L. L., 224Stanchescu, Y. V., 445Stancu, A., 504, 506, 563, 566Stanley, R. P., 399Starr, R., 150, 151Stavrakas, N. M., 70, 78Steenaerts, P., 264Steiger, W., 104Steiner, J., 223, 319Stepanov, V. N., 628Sternberg, S. R., 151Stoka, M., 261Stoker, J. J., 125, 454Stoyan, D., 151Strantzen, J. B., 168Straszewicz, S., 19Straus, E. G., 515Strausz, R., 8Study, E., 540Sturmfels, B., 154Su, B., 320, 628Süss, W., 309, 370, 378, 386, 438, 462, 463, 472,

578Sulanke, R., 558Sung, C. H., 90Szarek, S. J., 381Szegö, G., 391Sz.-Nagy, B. v., 223, 417Sz.-Nagy, G. v., 320Szwiec, C., 155

Talenti, G., 391Tam, B. S., 90Tani, M., 310Tanno, S., 188Teissier, B., 224, 398Temam, R., 150Tennison, R. L., 170Teufel, E., 527The Lai, Ph., 70Thomas, C., 399Thompson, A. C., xvi, 69, 391, 536, 565, 580, 582Tichy, R., 72



ExcerptIndex


Author index 727

Tietze, H., 15Tiwary, H. R., 154Tölke, J., xviTolstonogov, A. A., 70Tomczak-Jaegermann, N., 594Travaglini, G., xviiTreibergs, A., 464, 487Trudinger, N. S., 228, 399, 478, 558Tsang, A., 622Tsirelson, B. S., 310Tsolomitis, A., 381, 594Türk, I., 227Tverberg, H., 8Tweddle, I., 154Tzanaki, E., 154

Uhrin, B., 379, 380Umanskiy, V., 506Urbański, R., 70, 151, 167, 168, 170, 288

Valdimarsson, S. I., 597Valentine, F. A., xv, 10, 15Valettas, P., 606Valette, G., 60, 152van Heijenoort, J., 125van Schaftingen, J., 541Vanderwerff, J. D., xvi, 31Vanhecke, L., 223Varberg, D. E., xi, xvi, 31Vassallo, S., 511Vidal Abascal, E., 223Vilenkin, N. J., 627Villa, R., 381Vincensini, P., 153Viro, O., 340Vitale, R. A., 71, 73, 152, 154, 228, 308, 310, 381,

421, 558Vlasov, L. P., 10Voiculescu, D., 153, 381Volčič, A., 511, 540, 541Volkov, Yu. A., 387, 391, 486Volland, W., 338Vodop’yanov, S. K., 487Vritsiou, B.–H., 565, 569, 606

Waksman, Z., 90Wallen, L. J., 389Walter, R., 127, 132Wan, X., 586Wang, G., 583, 609Wang, T., 610, 611, 622Wang, Wei, 559, 587Wang, Weidong, 516, 559, 579, 585, 586Wang, X.–J., 228, 462, 478, 500, 506, 558Wang, Y., 399Wannerer, T., 228, 361, 365, 367, 368, 619Weberndorfer, M., 543, 601, 608, 609

Webster, R. J., xv, 71–73, 540Wegmann, R., 151Wegner, B., 188Wei, D., 559Weibel, Ch., 154Weil, W., 60, 112, 131, 152, 154, 167, 168, 188,

201, 204–207, 224, 227, 261–265, 273,287–290, 295, 308, 311, 312, 361, 362, 386,463, 473, 477, 478, 491, 495–497, 510, 529,532, 571, 582, 585

Weis, S., 90Weiss, A. Ivić, xviiWeiss, G., 623Weiss, M. L., xvWeiss, V., 236Weissbach, B., 201Weisshaupt, H., 72Wellerding, A., 71, 439Wenger, R., 8Wenzel, W., 73Werner, E., 525, 552, 556, 558, 561–563, 566, 595Wets, R. J.–B., 19, 69Weyl, H., 223, 439Wieacker, J. A., 137, 138, 161, 167, 168Wiernowolski, M., 168Wiesler, H., 19Wijsman, R. A., 69Wills, J. M., xvi, 264, 310, 389, 392, 409, 506Wills, M. D., 68Willson, S. J., 418Winternitz, A., 553Wintgen, P., 236Wintner, A., 125Witsenhausen, H. S., 205Wolff, P., 606Wrase, D., 409Wu, D., 587, 593Wu, H., 125Wulff, G., 411Wunderlich, W., 201

Xiang, Y., 559Xiao, J., 613Xiong, B., 580Xiong, G., 418, 506, 535, 576, 578

Yang, C., 569Yang, D., 389, 495–497, 500, 504–506, 540, 558,

568, 575–577, 591, 592, 595, 600, 601, 603,605, 607, 610–613

Yano, K., 310Yaskin, V., xvi, 473, 576, 582, 583, 586Yaskina, M., 473, 582, 586Yau, S.–T., 462Ye, D., 556, 559Yost, D., 169Yu, W., 577, 580, 587, 593



ExcerptIndex


728 Author index

Yuan, J., 514, 516, 579, 582, 583, 585, 595,606

Yuan, S., 514, 582, 583

Zähle, M., 127, 130, 132, 217, 226, 227, 235, 236,261, 264

Zaguskin, V. L., 594Zajı́ček, L., 31, 89, 103Zalgaller, V. A., 10, 15, 91, 103, 379, 390, 398Zamfirescu, T., xvi, 133, 134, 137, 152, 153Zarichnyi, M. M., 70Zhang, G., 188, 289, 294, 389, 418, 471, 477,

495–497, 500, 504–506, 510, 534, 535, 540,558, 567, 568, 573–577, 582, 584–586, 591,592, 595, 600, 601, 603, 605, 607, 610–613

Zhao, C.–J., 393, 512, 516, 559, 578, 579, 583Zhao, G., 543Zhao, L., 516

Zhong, X., 392Zhou, F., 478Zhou, J., 418, 559, 569Zhu, B., 559Zhu, G., 569Zhu, H., 583Zhu, M., 506Zhu, X., 399, 559, 583, 586Zhu, X.–Y., 583Zhuang, X., 151Ziegel, J., 321Ziegler, G. M., xvi, 8, 112Zinn, J., 154Živaljević, R. T., 73Zong, C., xviZvavitch, A., 204, 511, 543, 565, 580, 583, 585Zymonopoulou, M., 582, 583, 586



ExcerptIndex


Subject index

a-type, 109abstract duality, 43additive, 172, 329affine

combination, xixenergy, 610function, 20hull, xixisoperimetric inequality, 545normal vector, 544perimeter, 545Pólya–Szegö principle, 613quermassintegral, 515Sobolev inequality, 610surface area, 544, 545

affinely independent, xixAleksandrov body, 411Aleksandrov–Fenchel inequality, 393Aleksandrov–Fenchel–Jessen theorem,

448Aleksandrov’s projection theorem, 449Alesker’s irreducibility theorem, 365analogous (polytopes), 110antisummand, 139, 170area centroid, 314area measure, 214, 215Asplund sum, 517associated convex function, 51asymmetric Lp affine energy, 613asymmetric Lp zonotope, 609asymmetry class, 170, 472asymmetry function, 170

Baire space, 132ball

closed, xxiopen, xxi

Ball–Barthe inequality, 601Banach–Mazur distance, 589barycentre function, 19

barycentric coordinates, 6Binet ellipsoid, 590Blaschke

addition, 459, 463body, 463diagram, 387–Groemer inequality, 539–Minkowski homomorphism, 460–Santaló inequality, 548, 563

Lp, 568selection theorem, 63sum, 459

Blaschke’s rolling theorem, 164body of constant width, 140Bonnesen inequality, 388brightness function, 529Brunn–Minkowski theorem, 369Brunn–Minkowski theory

Lp, 489dual, 507

Busemann intersection inequality, 580Busemann–Petty problem, 583Busemann random simplex inequality, 539

canal class, 440cancellation law, 48, 139cap, 17, 91

body, 87covering theorem, 91

Carathéodory’s theorem, 3category

first, 132Cauchy’s surface area formula, 301centred support function, 262centroaffine surface area, 555centroid, 314centroid body

p-, 567Orlicz, 568

characteristic function, xx, 329



ExcerptIndex


730 Subject index

Chebyshev set, 10Choquet simplex, 532chord power integral, 534Christoffel’s problem, 470, 472circumball, 142circumradius, 142class reduction, 576closed

ball, xxiconvex function, 36convex hull, 6halfspace, xxsegment, xix

combinationaffine, xixconvex, 1linear, xixpositive, 2

commonantisummand, 467summand, 467support plane, 103

concaveα-, 520function, 20

cone-volume measure, 501cone-volume probability measure, 495conjugate face, 75conjugate function, 37constant width, 140continuous convex set, 59convex, 1

body, 8combination, 1cone, 1floating body, 561function, 19hull, 2

convolution body, 381, 561covariogram, 528Crofton’s intersection formula, 245cross covariogram, 530cross measure, 607current representation, 217curvature

centroid, 314function, 545image, 547measure, 214

cyclically monotonic, 30cylinder, 341

diameter, xxi, 49difference body, 140, 529

inequality, 530

dilatate, xxidilatation, xxidimension, 7direct sum, xxdirect summand, 156directly indecomposable, 156discriminant

mixed, 322dissection, 346distance, xxidual

Lp John ellipsoid, 593affine quermassintegral, 515Brunn–Minkowski theory, 507cone, 35harmonic quermassintegral, 514mixed volume, 508quermassintegral, 508

Dupin indicatrix, 126

economic cap covering, 104edge, 105effective domain, 20ellipsoid

Lp John, 592dual, 593

Binet, 590John, 587Legendre, 590Loewner, 587LYZ-, 591

elliptic type, 547p, 555

epigraph, 20equidissectable, 346Euler

characteristic, 214, 230point, 127relation, 337-type relation, 339

Euler’s theorem, 115even measure, 192even valuation, 353exceptional, 103exposed

r-, 76normal vector, 85support plane, 85

r-skeleton, 76face, 75normal vector, 85point, 18support plane, 85

exterior normal vector, 11, 81external angle, 109



ExcerptIndex


Subject index 731

extreme, 85(K1, . . . , Kn−1), 87r-, 76

normal vector, 85support plane, 85

normal vector, 85support plane, 85point, 16ray, 17

face, 16i-, 74-function, 78

facet, 74, 105Fenchel conjugate, 40first category, 132first fundamental form, 114flag measure, 227flat, xxfloating body, 561flotation body, 560form body, 386formula

Cauchy’s surface area, 301Crofton’s intersection, 245Kubota’s, 301principal kinematic, 239Steiner, 208

local, 213fully additive, 330function

affine, 20barycentre, 19characteristic, xxconcave, 20conjugate, 37convex, 19gauge, 53indicator, xx, 21support, 44

gauge function, 53gauge transform, 42, 55Gauss map, 113Gauss–Kronecker curvature, 115general Brunn–Minkowski theorem,

406general relative position, 474generalizedM-body, 191curvature measure, 213principal curvature, 129triangle body, 201zonoid, 195

generated Minkowski class, 190

generating measure, 193generating set, 160, 167generic, 132Geometric Barthe inequality, 597Geometric Brascamp–Lieb inequality, 597geometric convex function, 41geometric Minkowski combination, 494geominimal surface area, 549gradient, 26Grassmannian, xxiiGrothendieck group, 172

half-flat, xxhalf-open segment, xxHanner polytope, 564harmonic

p-combination, 512Blaschke addition, 513Blaschke linear combination, 513quermassintegral, 514

Hausdorffasymmetry index, 390distance, 61measure, xximetric, 61, 68

hedgehog, 526Helly’s theorem, 4hérisson, 526Hessian matrix, 28Hessian measure, 228homogeneous, 341homothet, xxihomothetic, xxi

positively, xxihomothety, xxihull

affine, xixclosed convex, 6convex, 2linear, xixpositive, 2

hyperbolic polynomial, 323hyperbolicity cone, 323hyperplane, xxhyperspace, 70

improper rigid motion, xxiiinclusion measure, 418inclusion–exclusion principle, 330indecomposable, 164indecomposable pair, 171independent

affinely, xixindex function, 231indicator function, xx, 21



ExcerptIndex


732 Subject index

inequalityLp Busemann–Petty centroid, 568affine Lp Sobolev, 610, 611affine Sobolev, 610Aleksandrov–Fenchel, 393Ball–Barthe, 601Blaschke–Groemer, 539Blaschke–Santaló, 548, 563Bonnesen, 388Busemann intersection, 580Busemann random simplex, 539difference body, 530Geometric Barthe, 597Geometric Brascamp–Lieb, 597geominimal surface area, 549Hölder’s for integrals, 509isodiametric, 382isoperimetric, 376, 382Jensen, 20Kneser–Süss, 460Minkowski’s, 382

first, 382for integrals, 509second, 382

Petty projection, 572affine, 579

Prékopa–Leindler, 373reverse Brunn–Minkowski, 380Rogers–Shephard, 530Urysohn, 382Zhang projection, 573

infimal convolution, 39inner radius, 143inradius

relative, 148internal, 7intersection body, 580

Lp, 581mixed, 583

intersectional family, 330intrinsic (r + 1)-moment, 314intrinsic volume, 208, 214irreducible, 169isodiametric inequality, 382isoperimetric inequality, 376, 382

affine, 545isotropic

constant, 605measure, 595

normalized, 596position, 604

Jensen’s inequality, 20Jessen radius of curvature, 125John ellipsoid, 587

John position, 596John’s theorem, 587

Kirchberger’s theorem, 14Klain

embedding, 356function, 356map, 356

Kneser–Süss inequality, 460Krein–Milman theorem, 18Kubota’s integral recursion, 301

Lφ affine surface area, 556Lψ affine surface area, 557Ln-star, 511Lp

addition, 490Brunn–Minkowski theory, 489John ellipsoid, 592Minkowski problem, 498Minkowski valuation, 617dual affine surface area, 559dual geominimal surface area, 559harmonic Blaschke addition, 513metric, 71surface area measure, 494zonoid, 606

-norm, 602lattice endomorphism, 34Legendre

ellipsoid, 590transform, 40transformation, 40

length measure, 465Lévy–Prokhorov distance, 480lift zonoid, 205line-free, 16lineality space, 16linear

combination, xixhull, xixparameter system, 542

linearity direction, 25local parallel set, 209local parallel volume, 121local Steiner formula, 213locally determined, 215locally embeddable, 157locally similar (polytopes), 110Loewner ellipsoid, 587Loewner position, 596log-concave function, 517lower curvature, 113lower radius of curvature, 113lower semi-continuous hull, 38



ExcerptIndex


Subject index 733

LYZ body, 611LYZ ellipsoid, 591

M-addition, 495maximal common summand, 171McMullen decomposition, 341meagre, 132mean

curvature, 115dual affine quermassintegral, 516section body, 265, 477width, 50, 173, 297, 519

measurable, 209measure

Hausdorff, xxiof non-convexity, 144

metric entropy, 72metric projection, 9metric tensor, 316minimal common anti-summand, 170minimal pair, 170, 472Minkowski

addition, 48additive, 48, 172class, 189difference, 146endomorphism, 177functional, 53, 208linear, 172problem

Lp, 498subtraction, 146symmetrization, 181tensor, 317valuation, 364, 586, 616

Lp, 617Minkowski’s

existence theorem, 455inequality, 382theorem, 17

Minkowskian integral formulae, 297, 309mixed

p-affine surface area, 559affine surface area, 548area measure, 279, 280body, 262, 463discriminant, 124, 322integral, 521intersection body, 583moment tensor, 321moment vector, 313projection body, 570quermassintegral, 427valuation, 345volume, 277, 280

width integral, 513moment body, 566

asymmetric Lp, 568moment matrix, 589moment vector, 312multiplier transformation, 180

nearest-point map, 9normal, 125

bundle, 127, 132, 209cone, 81point, 127vector, xx, 106

normalized convex body, 164normalized isotropic measure, 596

o-symmetric, xxobverse, 42odd valuation, 353open ball, xxiorder cancellation law, 139order isomorphism, 34Orlicz

centroid body, 568projection body, 575sum, 496zonotope, 609

outernormal vector, 11, 81parallel body, 128, 208unit normal vector, 106volume ratio, 599

p-addition, 490affine surface area, 554curvature function, 554geominimal surface area, 555linear combination, 490scalar multiplication, 490sum, 490tangential body, 86

parallel bodyinner, 148outer, 148

parallel chord movement, 543perfect face, 84Petty projection inequality, 572

affine, 579polar

Lp addition, 512body, 32curvature image, 546projection body, 570set, 32



ExcerptIndex


734 Subject index

polarization, 325polarization formula, 277polynomial

hyperbolic, 323valuation, 363

polytope, 3, 104porous, 136position, 596positive

basis, 15combination, 2hull, 2reach, 224

positively homogeneous, 24positively homothetic, xxiPrékopa–Leindler inequality, 373principal curvature, 114, 127

generalized, 129principal kinematic formula, 239principal radius of curvature, 116projection body, 302, 569

Lp, 575asymmetric, 575polar, 575

Lp mixed, 587Orlicz, 575

projection generating measure, 308proper

(convex function), 19(face), 74(rigid motion), xxiiseparation, 12

quasi-concave, 520quermassintegral, 208, 213quermassvector, 314

r-singular, 83r-skeleton, 76radial

p-combination, 508pth mean body, 574addition, 507Blaschke linear combination, 509Blaschke–Minkowski homomorphism,

512Blaschke sum, 509function, 57linear combination, 507map, 89metric, 507sum, 507valuation, 620

Radon’s theorem, 3rational homogeneous, 341ray, xxrecession cone, 16reduced pair, 170, 472reducible, 169regular, 83

normal vector, 87supporting halfspace, 84

relativeboundary, xxindecomposability, 171interior, xx

residual, 132reverse

Brunn–Minkowski inequality, 380isoperimetric inequality, 599second fundamental form, 116spherical image, 88

map, 88Weingarten map, 116

rigid motion, xxiRogers–Shephard inequality, 530rolling theorem, 164, 168roots of Steiner polynomial, 224rotation, xxiirotor, 189

Santaló point, 546Santaló region, 562second fundamental form, 114semiaxis function, 59separate, 12shadow system, 542shaking, 539Shapley–Folkman lemma, 141Shapley–Folkman–Starr theorem, 143shearing, 347similarity, xxiisimple

(polytope), 109(valuation), 346

simplex, 3simplicial, 109singular relative position, 251slicing problem, 606sliding freely, 156smooth, 83smooth valuation, 366Sobolev distance, 71special position, 238, 240specific curvature, 274



ExcerptIndex


Subject index 735

sphericalharmonic, 623image, 88

map, 88, 113Laplace operator, 119

stable convex body, 78star body, 57, 507star duality, 507starshaped, 57Steiner

ball, 421formula, 208point, 50, 315, 319

abstract, 181polynomial, 208

roots, 224symmetral, 536symmetrization, 536

Steinitz’s theorem, 14Straszewicz’s theorem, 18strict separation, 12strict subspace concentration inequality,

504strictly convex, 87strong separation, 12strongly isomorphic, 109subdifferential, 29subgradient, 29sublevel set, 20sublinear, 24subspace concentration condition, 504summand, 139, 147, 156sup-convolution, 517support, 11

cone, 81element, 59, 127, 209function, 44, 518measure, 213number, 276plane, xx, 11, 45set, 45vector, 293

supporting halfspace, 11, 45surface area measure, 214, 215

Lp, 494surface body, 562symmetric difference metric, 71

T-equidissectable, 346tangent space, 112tangential body, 86tangential radius of curvature, 126

telescoping, 443theorem

Aleksandrov’s projection, 449Aleksandrov–Fenchel–Jessen, 448Alesker’s irreducibility, 365Blaschke selection, 63Brunn–Minkowski, 369

general, 406Carathéodory, 3Euler, 115Groemer’s extension, 338Helly, 4John, 587Kirchberger, 14Krein–Milman, 18Minkowski, 17Minkowski’s existence, 455Radon, 3

coloured, 5Shapley–Folkman–Starr, 143Steinitz, 14Straszewicz, 18Tverberg, 8

thickness, 49touching cone, 85translate, xxitranslation, xxi

invariant, 341mixture, 262

triangle body, 201type

a-, 109typical, 132

umbilical point, 484unit ball, xxiunit sphere, xxiuniversal approximating class, 186universal convex body, 200upper

curvature, 113level set, 520radius of curvature, 113

Urysohn’s inequality, 382

valuation, 172, 329vertex, 105vertical (halfspace), 36virtual polytope, 526volume functional, 57volume product, 563



ExcerptIndex


736 Subject index

volume ratio, 597inner, 597outer, 599

weaklycontinuous, 348monotonic, 179positive measure, 179

weighted floating body, 562Weingarten map, 113width, 49

constant, 140function, 49, 140

integral, 514Wills functional, 310Wulff shape, 411, 418

Zhang projection inequality, 573zonal signed measure, 177zonoid, 191, 192

Lp, 606equation, 203generalized, 195

zonotope, 191antisymmetric Lp, 609Orlicz, 609



ExcerptIndex


http://www: cambridge: org:

9781107601017:

notation index - cambridge university...

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