Notation Index

Excerpt Index

Cambridge University Press 978-1-107-60101-7 - Encyclopedia of Mathematics and its Applications: Convex Bodies: The Brunn–Minkowski Theory: Second Expanded Edition Rolf Schneider Index More information

Notation index

+∞, 490 { f ≤ α},19 +ϕ, 496 f ∗ μ, 177 +p, 490 f g, 517 ∗ −A,xx f ,37 • A + B,xx f ,41 ◦ A − B,xx f , 523 A | E, xxii 1A,xx A ÷ B, 146 IIx, 114 A ⊕ B,xx Ix, 114 A ∨ B,33 A(n, k), xxii • A , 329 Aρ(K,β), 121 ∗ A C ,35 ,42 ◦ · K ,32 a(K, ), 59 ﬀ K, 507 a ,xix Kρ, 209 B(K), 421 [L, L], 240 B(z,ρ), xxi n [·, ·], xix B , xxi #, 459 B0(z,ρ), xxi & B • , 336 (X), 209 B(K), 590 ·K ,53 bij, 116 ≈S , 347 bd, xx ≈T , 347 + n− f¯, xxii C (S 1), 552 n−1 · , 490 C(S ), 177 p ∞ ηη, 251 C , 113 k f , 611 C , 113 k ·, ·,xix C+, 115 C (Rn), 522 ν ∗ μ, 460 α C (K, ·), 214 ⊕ , 495 m M Cvx(Rn), 40 II , 116 u Cvx (Rn), 41 ⊂,xx 0 CK, 547 )+ , 512 p C(X), 69 ) #, 513 Cn,60 ) F,75 CCn,42 !+, 507 CCn,33 ! o #, 509 C n,35 { f <α},19 c(K), 314 { f = α},19 c(K,β,u), 234

716 Notation index

cρ(K,η,x), 231 Gδ, 132 cl, xx Gp(K), 555 conv, 2 Gn, xxii χ, 214, 230 GL(n), xxii · D(K), xxi, 49 g(K, ), 53 D(·,...,·), 124, 322 gK , 528 Γ D (x ,...,x ), 303 φK, 568 k 1 k Γ DK, 140, 529 pK, 567 Γ−pK, 575 Du f , 610 d(A, x), xxi γ(F, P), 109 γn, 602 d(K, K ), 250 d(K, ·), 26 γ(F, F , P, P ), 240 · dBM, 589 H(K, ), 45 − diam, xxi H (K, ·), 45 + dim, 7 Hu,α,xx − dom, 20 Hu,α,xx δ,61 H j, 115 Q δ ,71 Hu,α,xx δS ,71 Hess, 28 δD,71 Hk, xxi · δp,71 h(K, ), 44 ∗ · δw,71 h (K, ), 262 δ˜, 507 ∞ IA , xx, 21 Δ, 119 + Ip , 582 Δ (K), 49 Ip, 581 ΔS , 119 IK, 580 ∂,29 i(K, q, u), 234

EpK, 592, 593 int, xx E(K), 75 J,42 EL(K), 587 j(K, q, x), 231 E (K), 587 J Ks, 546 E (K), 75 i Ku, 302 Ep( f ), 610 K f ,51 eij, 116 K[δ], 560 epi, 20 Kn s,1, 164 exp, 18 Kn s , 164 extr, 17 K(A), 8 ext, 17 Kn,8 expr,76 Kn(r, R), 91 extr,76 Kn c , 546 ηr,79 Kn n ,8 F(K, ·), 45 Kn (o),32 F (K), 74 Kn (os), 463 F n Kn , 545 os, 495 F n Kn c , 546 o ,32 F n K (o), 545 C , 440 Fi(K), 74 Kn(A), 8 f (K, ·), 545 Kn , 290 ! os Φi(K), 515 ki(x, u), 129 Φi(K), 515 κn, xxi Φr,s k , 317 κi(x, t), 113 Φm(K, ·), 216 κs(x, t), 113 G(K), 549 L(F), 240 G(n, k), xxii LK , 605

Notation index 717

n−1 n L2(S ), 623 Q , 523 n LC(R ), 517 qr(K), 314 L ,40 R(A, x), 9 L(K), 590 RpK, 574 lin , xix Rn,xix (K), 602 R¯ ,19 ΛK, 546 rec, 16 Λm(K, ·), 216 regn, 87 λF , 240 relbd, xx

Mk(K), 265 relint, xx M(Sn−1), 177 ρ(A), 143 − M(Sn 1, p), 177 ρi(x, t), 113 MK, 566 ρs(x, t), 113 MVal, 364 S (K, ·), 81 MValm, 364 S (·,...,·, ·), 279, 280 n M , 615 S m(K, ·), 214 m(L ⊂ K), 418 SH K, 536 μ, xxii, 237 SK , 465 μρ(K, ·), 209 SO(n), xxii μk, xxii, 237 SL(n), xxii N(K, F), 83 SV, 348 S N(K, x), 81 (K), 163 Sn N(P), 348 c , 546 Sn Nor K, 127, 209 o, 57, 507 Sn−1, xxi Nor , 59 S m, 623 ν, xxii, 237 s(K), 50 ν , xxii, 237 k s , 117 ∇, 26, 47 j s(K), 546 o,xix Σ, 209 Ω(K), 544, 545 σ, xxi, 469 Ω φ, 556 σ(K,β), 88 Ω (K ,...,K ), 559 p 1 n T(K, M), 262 Ω! −p(K), 559 T(K, u), 85 Ωp(K), 554 T(u1,...,un−1), 304 ωn, xxi TxK, 112 P(K, ·), 81 Θm(K,η), 212 projE , xxii τ,xix PK, 581 τ(K,ω), 88 P (A), 348 U(K), 91 n P , 8, 104 U(S ), 330 Pn • n, 8, 104 U (S ), 332 p˜, 325 Uij, 401 p(A, x), 9 U(S ), 332 pr(K), 314 u(A, x), 9 pos, 2 uK , 83, 88, 113 ΠK, 302, 569 V(·,...,·), 277, 280 Π f , 611 V1(K, L), 286 Π◦K, 570 VK (ω), 501 ΠφK, 575 V j(K), 208, 214 ΠpK, 575 Vp(K, L), 492 Ψr, 316 V(i), 406 Ψm(K, ·), 216 V(i)(K, L), 381 Q, 316 Vij, 421

718 Notation index

VK , 495 W˘ i, 514 ! V, 508 Wu, 116 ! ) Vi, 508 Wi(K), 514 ∞ ! Val , 366 Wn−i, 508 ∞ Valm , 366 w(K), 50, 297 Val, 353 w(K, ·), 49 + Valm, 353 − xK ,88 Val , 353 (k) m Ξm (K, ·), 227 ValG, 367 ξK , 115 Vn, 547 Υ Vn ,48 c , 547 (k) , 303 Zφ(Λ), 609 + Λ vert, 105 Zp ( ), 609 Z n vr(K), 597 s , 341 z(·,...,·), 313 Wi(K), 208, 213 zr+1(K), 312 Wi(·, ·), 427 Wx, 113

Author index

Abardia, J., 367, 580, 622 Baddeley, A. J., 236 Aczel,´ J., 349 Baebler, T., 390 Ader,O.B.,589 Bair, J., 15, 19, 153, 154, 171 Adiprasito, K., 137 Balashov, M. V., 167 Ahrens, I., 152 Ball, K., 374, 380, 523, 532, 573, 577, 588, 589, Alberti, G., 89 597, 599, 600, 602, 604 Aleksandrov, A. D., xi, xvi, 31, 58, 60, 89, 90, Bambah,R.P.,565 112, 127, 131, 202, 226, 274, 286, 287, 295, Banchoﬀ, T. F., 235 309, 327, 385, 398, 408, 409, 417, 438, 439, Bandle, C., 391 454, 461, 464, 472, 477, 486–488, 498, 505, Bandt, Ch., 72, 73 628 Bangert, V., 31, 274 Alesker, S., 60, 201, 206, 295, 361, 363, 365–368, Bantegnie, R., 72 398, 578 Bapat, R. B., 328 Alexander, R., 205 Baraki, G., 73 Alexandrov, V., 462 Bar´ any,´ I., 8, 104, 112, 137, 462, 558, 562 Alfsen, E. M., 18 Barthe, F., 377, 381, 525, 565, 598–600, 602 Allendoerfer, C. B., 223 Barthel, W., 390, 391, 580 Alonso-Gutierrez,´ D., 381, 578, 606, 609, Barvinok, A., xv, 594 614 Bastero, J., 380, 594, 606, 614 Batson, R. G., 19 Alvino, A., 390 Bauer, C., 170, 439, 454, 472 Ambartzumian, R. V., 204 Bauer, H., 18, 209 Ambrosio, L., 89 Baum, D., 59 Ambrus, G., 541 Bayen, T., 201 Anderson, R. D., 89 Bazylevych, L. E., 70 Andrews, B., 504, 506, 558 Beer, G. A., 69 Anikonov, Yu. E., 628 Behrend, F., 589, 593, 596, 600, 603 Arnold, R., 71, 320, 428, 439, 487 Bensimon, D., 78 Arocha, J. J., 8 Benson, R. V., xv, 393 Arrow, K. J., 150, 151 Benyamini, Y., 391 Artstein, Z., 150, 152, 154 Berck, G., 581 Artstein-Avidan, S., 40, 42, 43, 182, 518, 524, Berg, Ch., 181, 186, 470, 628 525 Berger, M., xv Ash, R. B., 209 Bernig, A., 228, 236, 261, 366–368, 578, 580, 622 Asplund, E., 32, 76 Bernstein,ˇ D. N., 398 Assouad, P., 205 Bernstein, J., 361, 367 Atkinson, K., 623 Bernues,´ J., 380, 606, 614 Auneau-Cognacq, J., 265, 321 Besicovitch, A. S., 89, 103 Averkov, G., 529 Betke, U., 202, 204, 311, 386 Avriel, M., 520 Bettinger, W., 390

720 Author index

Bianchi, G., 31, 529, 532, 540, 541 Burton, G. R., 80, 168, 202 Bianchini, C., 532, 543 Busemann, H., xvi, 31, 125, 127, 131, 274, 327, Biehl, Th., 540 390, 391, 398, 454, 462, 487, 515, 536, 539, Bisztriczky, T., xvii, 541, 569 580, 583 Bjorck,¨ G., 77 Caﬀarelli, L. A., 380, 462 Blackwell, D., 205 Campi, S., 379, 463, 473, 540, 541, 543, 566, 568, Blagojevic,´ P. V. M., 8 569, 608, 628 Blaschke, W., xii, xv, 69, 70, 126, 164, 181, Cannarsa, P., 89 201–204, 261, 308, 320, 361, 378, 388, 463, Caratheodory,´ C., 3, 8, 540 472, 473, 538, 540, 541, 543, 545, 546, 548, Carlen, E. A., 605 550, 560, 591, 595, 628 Cascos, I., 206 Blind, R., 487 Cassels, J. W. S., 150 Blumenthal, L. M., 72 Chai, Y. D., 516 Bobkov, S. G., 520, 522, 523, 525, 614 Chakerian, G. D., 201, 203, 393, 417, 454, 463, Bohm,¨ J., 338 477, 530, 532, 534, 586 Bor¨ oczky,¨ K., 462 Chang, S.–Y. A., 399 Bor¨ oczky,¨ K. J., 34, 380, 389, 393, 445, 495, 504, Chavel, I., 390, 391 506, 525, 534, 541, 558, 565, 566, 569, 576, Cheeger, J., 236 605 Chen, B., 295, 339, 361 Bokowski, J., 264, 310, 389 Chen, F., 569 Bol, G., 149, 151, 388, 389, 417, 432, 444, 628 Chen, L.–Y., 393 Bolker, E. D., 201–205 Chen, W., 506, 558 Boltyanski, V., xvi, 201, 418 Cheng, S.–Y., 462 Boman, J., 126 Chern, S. S., 125, 261, 309, 454 Bonnesen, T., xi, xv, xvi, 58, 60, 81, 90, 91, 155, Chernoﬀ, P. R., 628 188, 286, 287, 308, 320, 370, 378, 386, 388, Cheung, W.–S., 393, 418, 512, 559, 578, 579, 583, 393, 461 606 Bonnice, W., 15 Choquet, G., 18, 79, 202, 205 Borell, C., 380, 517, 520 Chou, K.–S., 462, 500, 506 Borowska, D., 167 Cianchi, A., 613 Borwein, J. M., xvi, 31, 144, 151 Cieslak,´ W., 628 Bose, R. C., 319, 320 Clack, R., 606 Botts, T., 9 Coifman, R. R., 623 Bourbaki, N., 15, 18, 79, 90 Colesanti, A., 31, 89, 228, 273, 379, 380, 463, Bourgain, J., 181, 203, 204, 378, 387, 486, 540, 519, 520, 522, 523, 525, 532, 540, 541, 543 541, 565, 585, 605, 628 Collier, J. B., 79 Bocek,ˇ L., 393 Colzani, L., xvii Bracho, J., 8 Cordero-Erausquin, D., 605 Brandolini, L., xvii Corson, H. H., 79 Brannen, N. S., 389, 578 Coupier, D., 541 Brascamp, H. J., 380 Coxeter, H. S. M., 202 Brehm, U., 236 Cressie, N., 154 Brickell, F., 545 Cuoghi, P., 380 Brocker,¨ L., 236, 367 Curtis,D.W.,70 Brøndsted, A., xvi, 19, 90, 112 Czipszer, J., 417 Bronshtein, E. M., 72, 73, 79, 172, 463 D’Agata, A., 151 Brooks, J. N., 168 Dalla, L., 79, 80, 541 Brown, A. L., 78 Danzer, L., 8, 15, 587 Brunn, H., xi, 377, 535 Dar, S., 379, 398 Buchta, C., 562 Das Gupta, S., 380 Budach, L., 236 Davis, C., 15 Bunt, L. N. H., 10 Davy, P. J., 321 Burago, Ju. D., 10, 15, 274, 379, 390, 398 Davydov, Yu., 541 Burchard, A., 540 de Rham, G., 137 Burckhardt, J. J., 202 Debrunner, H., 310

Author index 721

Debs, G., 78 Fillmore, J. R., 473, 628 Deicke, A., 545, 546 Firey, W. J., 59, 60, 80, 126, 167, 168, 188, 203, Delgado, J. A., 168 263, 273, 288, 310, 387, 393, 463, 470, 473, Dierolf, P., 69 477, 490, 497, 504, 510, 512, 513 Dinghas, A., 71, 152, 310, 378, 379, 388, 390, 417 Fish, A., 583 418, 444, 628 Fisher,J.C.,628 Diskant, V. I., 378, 384, 387, 389, 391, 409, 417, Flaherty, F. J., 236 445, 478, 482, 484, 486, 487 Flanders, H., 319, 389 Dolzmann, G., 552 Flatto, L., 101 Dor, L. E., 205 Fleury, B., 569 Dou, J., 506 Florentin, D., 43 Dreseviˇ c,´ M., 69 Florian, A., 71, 392 Duan, X., 516 Focke, J., 201, 628 Dubins, L. E., 19 Fogel, E., 154 Dubois, C., 154 Folkman, J., 141, 150 Dubuc, S., 380 Fourneau, R., 15, 154, 169, 533 Dudley, R., 32, 72 Fradelizi, M., 409, 525, 543, 564, 565 Dunkl, C. F., 179 Fragala,` I., 519, 520, 522, 523 Duporcq, E., 320 Franchi, B., 410 Dvoretzky, A., 565 Franz, G., 580 Dyer, M., 289, 312 Freiman, G. A., 445 Dziechcinska-Halamoda,´ Z., 155 Fresen, D., 563 ff Eckho , J., 8, 339 Fu, J. H. G., 31, 228, 236, 261, 262, 366–368 Edelstein, M., 79, 154 Fuglede, B., 19, 388, 391, 628 Efimow, N. W., 439 Fujiwara, M., 90, 201, 628 Eggleston, H. G., xv, 79, 80, 388, 532, 540 Fukuda, K., 154 Egorychev, G. P., 327 Fusco, N., 391, 392, 628 Eifler, L. Q., 78 Ekeland, I., 150 Gage, M. E., 388, 504, 506 Epelman, M. S., 90 Gale, D., 59, 167, 168 Evans, L. C., 31 Gallego, E., 367, 410, 527 Ewald, G., 60, 91, 103, 104, 152, 170, 429 Gallivan, S., 79 Gao, F., 228, 410 Fabila, R., 8 Gårding, L., 323, 327 Falconer, K. J., 628 Gardner, R. J., xvi, 79, 152, 290, 369, 374, 377, Faro Rivas, I., 188 379, 411, 418, 454, 463, 491, 495–497, 510, Fary,´ I., 287, 417, 462, 543, 561 511, 516, 532, 535, 539, 572, 574, 576, 582, Favard, J., 89, 90, 389, 393, 409, 410, 431, 438, 584, 586, 610 439, 444, 472 Gariepy, R. F., 31 Federer, H., 31, 89, 224, 226, 236, 261, 262, 272, 273, 390 Gates, J., 364 Fedotov, V. P., 79, 90, 310, 477, 497 Geivaerts, M., 152, 167, 168 Fejes Toth,´ L., 389, 392, 541, 558 Gensel, B., 201 Feller, W., 31, 127, 131 Geppert, H., 444, 526, 628 Fenchel, W., xi, xv, xvi, 58, 60, 81, 90, 91, 155, Gericke, H., 319, 410, 628 188, 223, 286, 287, 308, 320, 370, 378, 386, Ghandehari, M., 393 398, 409, 439, 454, 461 Giannopoulos, A., 409, 541, 565, 569, 594, 603, Feng, Y., 579 604, 606 Ferone, V., 390 Gine,´ E., 73, 154 Ferrari, F., 410 Glasauer, S., 104, 225, 227, 262 Ferrers, N. M., 320 Gluck, H., 462 Fesmire, S., 154 Godbersen, C., 532, 533 Figalli, A., 377, 383, 392 Godet-Thobie, Ch., 70 Figiel, T., 203, 391 Gortler,¨ H., 289, 628 Fillastre, F., 527 Goikhman, D. M., 463 Filliman, P., 202, 439 Goldberg, M., 201

722 Author index

Goodey, P. R., 168, 201, 202, 204, 206, 207, 227, Hartzoulaki, M., 409, 541 262, 265, 273, 287, 289, 308, 311, 438, 463, Hausdorﬀ, F., 68, 69 473, 477, 478, 529, 582, 584, 628 Hayashi, T., 320, 628 Goodman, A. W., 320 He, B., 506, 559, 576, 583, 587, 606 Goodman, J. E., xvii Heil, E., 69, 201, 298, 310, 389, 390, 393, 565 Gordon, Y., 203, 525, 565, 566, 594 Heine, R., 399 Go´zd´ z,´ S., 628 Helly, E., 4, 8 Gray, A., 223 Henk, M., 224, 506, 588 Green, J. W., 60, 603 Herglotz, G., 223, 463 Green, M., 225 Hernandez´ Cifre, M. A., 224, 225, 393, 440 Grinberg, E. L., 188, 471, 515, 582, 586 Hertel, E., 338 Gritzmann, P., 154, 202, 289, 312, 408, 409, Heveling, M., 224 462 Hilbert, D., 378, 398 Groemer, H., xvi, 132, 201, 264, 288, 292, 293, Hildenbrand, W., 69, 151, 205, 207 332, 334, 338–340, 376–378, 383, 386, 387, Hille, E., 58 391, 392, 421, 438, 539, 623, 628 Hinderer, W., 227, 361 Gromov, M., 52, 60, 378, 398, 612 Hiriart-Urruty, J.–B., 32 Gronchi, P., 379, 463, 473, 540, 541, 543, 566, Hirose, T., 69 568, 569, 608 Hobinger,¨ J., 536 Gross, W., 538, 540 Hormander,¨ L., xvi, 42, 70, 323, 327, 398 Gruber, P. M., xv, xvi, 31, 42, 71–73, 112, Hoﬀman, L. M., 73 136–138, 167, 411, 418, 533, 536, 540, 558, Holicky,´ P., 79 588, 594 Holmes, R. B., 15 Grunbaum,¨ B., xvi, 8, 15, 73, 79, 80, 112, 155, Holmes, R. D., 391 169, 181, 186, 463, 477, 532 Horn,B.K.P.,462 Grynkiewicz, D., 445 Howard, R., 202, 454, 558 Grzaslewicz,´ R., 171 Howe, R., 137 Grzybowski, J., 151, 167, 168, 170, 171, 472 Hu, C., 506 Guedon,´ O., 569 Hu, Y., 497, 566 Guan, B., 478 Huang, Q., 505, 583, 609 Guan, P., 399, 478 Huang, Y., 506 Guggenheimer, H., xvi, 487, 564 Huck, H., 309 Guilfoyle, B., 201 Hufnagel, A., 289, 312, 462 Guleryuz, O. G., 595 Hug, D., 89, 125, 127, 132, 152, 205, 224, 225, Guo, Q., 155 227, 228, 262, 273, 274, 290, 312, 321, 361, Gurvits, L., 327 363, 364, 379, 410, 454, 463, 482, 486, 491, Gustin, W., 600 495–497, 500, 510, 532, 552, 555, 564, 565, 594, 604 Haberl, C., 504, 557, 568, 575, 582, 585, 613, Hurwitz, A., 472, 628 620, 621 Husain, T., 154 Hadamard, J., 125 Hadwiger, H., xii, xv, 69, 151, 173, 181, 182, 202, Inzinger, R., 628 223, 235, 239, 261, 264, 310, 319, 320, 338, Iosevich, A., xvii 339, 341, 346–348, 360–363, 378, 379, 380, Ivaki, M. N., 506 387–390, 392, 393, 398, 417, 440, 444, 514, Ivanov,B.A.,104 540 Ivanov, G. E., 168 Hahn, F. H., 150, 151 Jacobs, K., 18 Hahn, M. G., 73, 154 Jaglom, I. M., xvi, 201 Halmos, P. R., 205 Jenkinson, J., 595 Halperin, D., 154 Jensen, E. B. V., 265, 321, 511 Hammer, P. C., 153, 188 Jerison, D., 380 Han, W., 623 Jessen, B., 31, 60, 125, 223, 286, 287, 409, 454, Hann, K., 125, 225 461 Haralick, R. M., 151 Jetter, M., 225 Hartenstine, D., 512 Jiang, J., 497 Hartman, P., 126 Jiang, M. Y., 506

Author index 723

Jimenez,´ C. H., 381, 589 Kuiper, N. H., 236 Jin, H. L., 155 Kuperberg, G., 565, 578 John, F., 588, 591 Kuppe, M., 236 Jonasson, J., 534 Kuratowski, C., 69 Jongmans, F., 18, 70, 72, 89, 154, 171 Kutateladze, S. S., 462, 463 Jourlin, M., 155 Lachand-Robert, T., 201 Juhnke, F., 588 Laczkovich, M., 79 Kaasalainen, M., 462 Lafontaine, J., 236 Kahn, J., 399 Lagarias,J.C.,327 Kakeya, S., 201, 628 Laget, B., 155 Kalinin, B. V., 274 Lamberg, L., 462 Kallay, M., 60, 112, 169, 171, 472 Langevin, R., 526 Kalman, J. A., 19 Larman, D., 8, 78–80, 90, 91, 103, 104, 477, 541, Kaluza, T., 151 562 Kameneckii, I. M., 201, 628 Lashof, R. K., 125 Kampf, J., 223 Last, G., 224 Karasev,¨ R. N., 8, 167 Laugwitz, D., 57, 587 Karavelas, M. I., 154 Lay, S. R., xv Karcher, H., 168 Lebesgue, H., 201 Katsnelson, V., 224, 225 Ledoux,M., 614 Katsurada, Y., 309 Lee, Y. S., 516 Kelly, P. J., xv Lehec, J., 525 Kendall, W. S., 151 Leichtweiß, K., xv, xvi, 59, 287, 327, 398, 438, Kenderov, P., 137 454, 487, 526, 543, 548, 550, 553, 560, 561, Khovanski˘ı, A. G., 327, 340, 363, 398 564, 589, 595 Kiderlen, M., 179, 182, 460, 463, 532 Leindler, L., 373 Kim, J., 566, 583 Leng, G., 380, 454, 506, 511, 513, 514, 516, 559, Kincses, J., 167 565, 569, 576, 579, 582, 583, 585, 586, 593, Kiselman, C. O., 126 595, 606, 609 Klain, D. A., 340, 355, 356, 360, 362, 364, 378 Lenz, H., 339, 587 462, 511, 540 Letac, G., 472 Klartag, B., 182, 328, 399, 519, 525, 540, 569, Lettl, G., 71, 72 594, 605 Levin, B. Ja., 472 Klee, V., xvi, 8, 15, 18, 19, 59, 78–80, 89, 103, Levitt, G., 526 136, 153, 155, 167, 272, 289, 339, 532 Levy,´ P., 205 Klein, E., 69 Lewis, D. R., 588, 594, 607 Klima, V., 137 Lewis, J. E., 152 Klingenberg, W., 201 Lewy, H., 462 Klotzler,¨ R., 201 Li, A.–J., 569, 603, 606 Kneser, H., 59, 370, 378, 463 Li, A.–M., 543 Kneser, M., 224 Li, D.–Y., 418 Knothe, H., 372, 378, 388, 393, 439, 628 Li, J., 399, 478 Knyazeva, M., 526 Li, K., 506, 576 Konig,¨ H., 586 Li, N., 559 Kohlmann, P., 236, 310, 487 Li, X., 511 Koldobsky, A., xvi, 576, 582–584, 586 Li, Y., 478, 506 Kone, H., 622 Liapounoﬀ, A. A., 205 Kopteva, N., 462 Lieb, E. H., 380 Koutrouﬁotis, D., 168, 487 Lin, C., 478 Krantz, S. G., 126 Lin, Y., 565 Krein, M., 18 Lindelof,¨ L., 385 Kropp, R., 227 Lindenstrauss, J., 181, 203–205, 378, 387, 391, Kruskal, J. B., 78 486, 540, 591, 628 Kubota, T., 319, 320, 409, 438, 472, 628 Lindquist, N. F., 202, 206 Kuhnel,¨ W., 236 Linhart, J., 202, 204

724 Author index

Linke, Ju. E., 73 McMullen, P., xi, xvi, xvii, 9, 59, 72, 73, 76, 112, Little, J. J., 462 131, 155, 165, 167–169, 171, 202, 204, 288, Litvak, A. E., 534, 594 289, 319, 321, 332, 339, 340, 346, 352, 353, Liu, L., 587 362, 366, 378, 461, 463, 533 Ljasenko,ˇ N. N., 154 Mecke, J., 151 Lonke, Y., 203, 582 Meier, Ch., 295, 319, 346 Lopez, M. A., 566 Meissner, E., 201, 319, 320, 628 Loritz, A., 439 Melzak, Z. A., 137 Lu,¨ S.–J., 586 Menger, K., 72 Lu, F., 506, 513, 516, 559, 583, 595 Meschkowski, H., 152 Lu, G., 410 Meyer, M., 381, 409, 438, 525, 530, 536, 541, Lu, J., 506 543, 556, 560, 562–566, 583, 594 Lu, Q., 506 Meyer, P., 223 Ludwig, M., 532, 554, 556–558, 568, 575, 582, Meyer, W., 112, 167, 169, 181 613–616, 618, 620–622 Michael, E., 69 Lutwak, E., xvii, 288, 389, 393, 454, 460, 463, Milka, A. D., 171 490, 495–498, 500, 504–507, 510, 512–516, Milman, D., 18 540, 546, 549–551, 554, 558, 559, 564, 567, Milman, E., 569 568, 570–572, 575–578, 580–582, 584–586, Milman, V., 33, 40–43, 181, 182, 203, 288, 380, 591, 592, 595, 600, 601, 603, 605, 607, 391, 398, 409, 518–520, 523–525, 540, 541, 610–613 565, 580, 591, 594, 603, 604, 606, 628 Minkowski, H., xi, 18, 90, 151, 155, 188, 286, Lv, S., 454, 511, 559, 579, 586, 595 287, 320, 377, 385, 386, 431, 432, 439, 453, Lyusternik, L. A., xvi, 390 455, 461, 463, 628 Ma, T.–Y., 586, 595 Minoda, T., 628 Ma, X.–N., 399, 478, 506 Miranda, C., 462 Macbeath, A. M., 73, 539 Molchanov, I., 150, 154, 206 Maehara, H., 160 Montejano, L., 8, 70, 167 Maggi, F., 377, 383, 392 Moore, J. D., 487 Mahler, K., 564, 565 Moreno, J. P., 167 Makai, E., 223 Mosler, K., 206 Makai, E., Jr, 462, 533, 541, 564, 566 Moszynska,´ M., xv, 507 Malkevitch, J., xvii Motzkin, T. S., 10 Mani-Levitska, P., 15, 79, 261, 331, 332, 339, 541 Muller,¨ C., 623 Mao, W., 583 Muller,¨ H. R., 320, 321 Marchaud, A., 535 Muller,¨ W., 236 Maresch, G., 609 Murner,¨ P., 202 Markessinis, E., 606 Naden´ ˆık, Z., 393, 477 Marti, J. T., xvi, 10, 15, 31, 89 Nadler, S., 70 Martin, K., 78 Nakajima, S., 320, 628 Martinez-Maure, Y., 488, 526 Naszodi,´ M., 589 Martini, H., 73, 201, 202, 418, 532, 541, 578, 582 Nazarov, F., 204, 565, 583 Mase, S., 154 Netuka, I., 137 Matheron, G., 69, 151, 168, 202, 205, 225, 235, Neveu, J., 240 236, 289, 308, 417, 529 Newman, D. J., 101 Matousek,ˇ J., 8 Neyman, A., 207 Matschke, B., 8 Nikliborc, W., 320 Matsumura, S., 409, 628 Nirenberg, L., 462 Matveev, V. S., 595 Nitsch, C., 390 Mazur, S., 89 Noll, D., 31, 132 McAllister, B. L., 69 O’Brien, R. C., 144, 151 McCann, R. J., 32, 378 Oda, T., 224 McClure, D. E., 558 Ohmann, D., 223, 379, 387, 388, 390, 410 McKinney, R. L., 15 Oishi, K., 628 McMinn, T. J., 103 Oliker, V. I., 454, 464, 486, 506

Author index 725

Oliveros, D., 8 Quinn, J., 70 Osher, S., 225 Redei,´ L., 561 Oshio, S., 417 Renyi,´ A., 558 Osserman, R., 388–391 Rademacher, H., 58 ´ Oudet, E., 201 Radon, J., 3, 8, 188 Overhagen, T., 310 Rådstrom,¨ H., 70 Pach, J., 462 Rajala, K., 392 Pajor, A., 541, 564, 580, 591, 594, 604, 606 Rataj, J., 227, 236, 265, 312 Pallaschke, D., 170, 288 Ratschek, H., 70 Palmon, O., 589 Rauch, J., 126, 168 Panina, G. Yu., 204, 293, 340, 488, 526 Reay,J.R.,15 Panov, A. A., 328 Redei,´ L., 543 Paouris, G., 310, 409, 541, 565, 569, 582, 583, Reidemeister, K., 89, 204 595, 606 Reisner, S., 381, 438, 525, 530, 536, 543, 560, Papaderou–Vogiatzaki, I., 263 564–566, 583 Papadimitrakis, M., 603 Reiter, H. B., 78 Papadopoulou, S., 78 Reitzner, M., 104, 112, 554, 556–558, 562 Parapatits, L., 361, 365, 463, 557, 579, 618, 619, Ren, D., 534 621 Renegar, J., 323, 327 Parks, H. R., 126 Renyi,´ A., 417 Pastor, J. A., 393 Resetnjak,ˇ Ju. G., 31, 487, 628 Pavlica, D., 103 Reuleaux, F., 201 Paya,´ R., 169 Ricker, W., 207 Pelczynski,´ 588 Rickert, N. W., 202, 205 Pena,˜ A., 380 Riesz, F., 59 Perissinaki, I., 594 Roberts, A. W., xi, xvi, 31 Perles, M. A., 235, 338 Rockafellar, R. T., xi, xv, 9, 19, 30–32, 41, Petermann, E., 393 42, 59 Petrov, F., 565 Rodr´ıguez, L., 526 Petty, C. M., 202, 203, 536, 546–549, 564, 567, Rogers, C. A., xii, 69, 79, 91, 103, 104, 169, 477, 569, 570, 572, 579, 581, 583, 591, 594, 603, 530, 532, 533, 541, 565 628 Romance, M., 594, 606 Pﬁefer, R. E., 539 Rosenberg, H., 526 Phelps, R. R., 18, 186 Rota, G.–C., 339, 364 Phillips, R. S., 58 Rotem, L., 43, 519, 520, 522–524 Pisier, G., 374, 380, 565 Rother, W., 261, 264 Pivovarov, P., 310, 569 Roy, A. K., 154 Pogorelov, A. V., xvi, 274, 461, 462, 471–473, Roy, N. M., 18 486, 487 Roy, S. N., 319, 320 Pollack, R., xvii Rubin, B., 584 Polovinkin, E. S., 160, 167 Rubinov, A. M., 463 Polya,´ G., 391 Rudelson, M., 603 Pompeiu, D., 68 Ryabogin, D., 204, 565, 580, 583, 585 Ponsiglione, M., 392 Sacksteder, R., 125 Pontryagin, L. S., 151, 181 Sah, C.–H., 360 Positsel’skii, E. D., 174, 181 Saint Pierre, J., 73, 181, 188 Pranger, W., 19 Saint Raymond, J., 564 Pratelli, A., 377, 383, 392, 628 Saks, M., 399 Prodromou, M., 558 Salani, P., 380 Protasov, V. Ju., 495 Salinas Mart´ınez, G., 393 Przesławski, K., 52, 60, 73, 287, 322, 339 Salinetti, G., 69 Przybycien,´ H., 151, 167 Salkowski, E., 543 Pucci, C., 31, 89 Sallee, G. T., 131, 151, 160, 166, 167, 169, 181, Pukhlikov, A. V., 340, 363 235, 332, 338–340 Qi, C., 559 Sandgren, L., 59

726 Author index

Sangwine-Yager, J. R., 223–226, 387, 389, 417, Si, L., 580, 595 439 Silverman, R., 169 Santalo,´ L. A., 223, 261, 310, 388, 393, 410, 546, Simon, U., 309, 310, 543 548 Slomka, B. A., 42, 43 Saor´ın, E., 224, 225, 379, 440 Smilansky, Z., 169 Saroglou, C., 541, 543, 578, 606 Soberon,´ P., 8 Sas, E., 541, 628 Solanes, G., 228, 367, 410, 527 Saskin,ˇ Ju. A., 18 Soltan, P. S., 418 Schaal, H., 201 Soltan, V., 532 Schechtman, G., 391, 602 Song, X., 535 Schirokow, A. P., 543 Sorger, H., 137 Schirokow, P. A., 543 Sorokin, V. A., 152 Schmidt, E., 378, 391 Spiegel, W., 69, 332, 346 Schmidt, K. D., 70 Stacho,´ L. L., 224 Schmitt, K.–A., 152 Stanchescu, Y. V., 445 Schmuckenschlager,¨ M., 381, 530, 561, 577, 602 Stancu, A., 504, 506, 563, 566 Schneider, R., xv, xvii, 34, 35, 60, 71–73, 78, 80, Stanley, R. P., 399 90, 104, 112, 126, 137, 144, 151, 155, 167, Starr, R., 150, 151 168, 170, 171, 179–182, 188, 190, 198, Stavrakas, N. M., 70, 78 200–203, 205, 206, 224, 226, 235, 236, Steenaerts, P., 264 261–264, 273, 274, 287, 288, 295, 309, 312, Steiger, W., 104 320, 321, 332, 338, 358, 362–364, 391, 399, Steiner, J., 223, 319 409, 410, 417, 421, 428, 438, 439, 444, 450, Stepanov, V. N., 628 454, 461, 463, 471–473, 477, 482, 484, Sternberg, S. R., 151 486–488, 497, 516, 529, 533, 535, 540, 545, Stoka, M., 261 558, 578, 582, 583, 594, 604, 628 Stoker, J. J., 125, 454 Scholtes, S., 170, 288, 472 Stoyan, D., 151 Schori, R. M., 70 Strantzen, J. B., 168 Schroder,¨ G., 70 Straszewicz, S., 19 Schrader, R., 236 Straus, E. G., 515 Schrocker,¨ H.–P., 594 Strausz, R., 8 Schurger,¨ K., 154 Study, E., 540 Schurmann,¨ A., 506 Sturmfels, B., 154 Schutt,¨ C., 525, 552, 556, 558, 561–563, 566 Su, B., 320, 628 Schuster, F. E., 179, 182, 200, 201, 206, 364–366, Suss,¨ W., 309, 370, 378, 386, 438, 462, 463, 472, 368, 461, 463, 512, 568, 575, 578, 579, 586, 578 588, 594, 601, 609, 613, 619, 628 Sulanke, R., 558 Schuster, R., 321, 363 Sung, C. H., 90 Schwarz, T., 137 Szarek, S. J., 381 Schwella, A., 409 Szego,¨ G., 391 Seeger, A., 32 Sz.-Nagy, B. v., 223, 417 Seeley, R. T., 119, 623 Sz.-Nagy, G. v., 320 Segal, A., 42, 43, 377, 379, 384 Szwiec, C., 155 Segura Gomis, S., 393 Talenti, G., 391 Sen’kin, E. P., 473 Tam, B. S., 90 Serra, J., 151 Tani, M., 310 Serra, O., 445 Tanno, S., 188 Shahin, J. K., 310 Teissier, B., 224, 398 Shapley, L., 141, 150 Temam, R., 150 Shen, C., 506 Tennison, R. L., 170 Shen, Y., 583 Teufel, E., 527 Sheng, W., 478, 558 The Lai, Ph., 70 Shephard, G. C., xi, xvi, 9, 15, 71, 72, 112, 152, Thomas, C., 399 164, 167–170, 181, 186, 202, 235, 319, 340, Thompson, A. C., xvi, 69, 391, 536, 565, 580, 582 399, 530, 532, 533, 541, 583, 628 Tichy, R., 72

Author index 727

Tietze, H., 15 Webster, R. J., xv, 71–73, 540 Tiwary, H. R., 154 Wegmann, R., 151 Tolke,¨ J., xvi Wegner, B., 188 Tolstonogov, A. A., 70 Wei, D., 559 Tomczak-Jaegermann, N., 594 Weibel, Ch., 154 Travaglini, G., xvii Weil, W., 60, 112, 131, 152, 154, 167, 168, 188, Treibergs, A., 464, 487 201, 204–207, 224, 227, 261–265, 273, Trudinger, N. S., 228, 399, 478, 558 287–290, 295, 308, 311, 312, 361, 362, 386, Tsang, A., 622 463, 473, 477, 478, 491, 495–497, 510, 529, Tsirelson, B. S., 310 532, 571, 582, 585 Tsolomitis, A., 381, 594 Weis, S., 90 Turk,¨ I., 227 Weiss, A. Ivic,´ xvii Tverberg, H., 8 Weiss, G., 623 Tweddle, I., 154 Weiss, M. L., xv Tzanaki, E., 154 Weiss, V., 236 Weissbach, B., 201 Uhrin, B., 379, 380 Weisshaupt, H., 72 Umanskiy, V., 506 Wellerding, A., 71, 439 Urbanski,´ R., 70, 151, 167, 168, 170, 288 Wenger, R., 8 Valdimarsson, S. I., 597 Wenzel, W., 73 Valentine, F. A., xv, 10, 15 Werner, E., 525, 552, 556, 558, 561–563, 566, 595 Valettas, P., 606 Wets,R.J.–B.,19,69 Valette, G., 60, 152 Weyl, H., 223, 439 van Heijenoort, J., 125 Wieacker, J. A., 137, 138, 161, 167, 168 van Schaftingen, J., 541 Wiernowolski, M., 168 Vanderwerff, J. D., xvi, 31 Wiesler, H., 19 Vanhecke, L., 223 Wijsman, R. A., 69 Varberg, D. E., xi, xvi, 31 Wills, J. M., xvi, 264, 310, 389, 392, 409, 506 Vassallo, S., 511 Wills, M. D., 68 Vidal Abascal, E., 223 Willson, S. J., 418 Vilenkin, N. J., 627 Winternitz, A., 553 Villa, R., 381 Wintgen, P., 236 Vincensini, P., 153 Wintner, A., 125 Viro, O., 340 Witsenhausen, H. S., 205 Vitale, R. A., 71, 73, 152, 154, 228, 308, 310, 381, Wolff, P., 606 421, 558 Wrase, D., 409 Vlasov, L. P., 10 Wu, D., 587, 593 Voiculescu, D., 153, 381 Wu, H., 125 Volciˇ c,ˇ A., 511, 540, 541 Wulff, G., 411 Volkov, Yu. A., 387, 391, 486 Wunderlich, W., 201 Volland, W., 338 Xiang, Y., 559 Vodop’yanov, S. K., 487 Xiao, J., 613 Vritsiou, B.–H., 565, 569, 606 Xiong, B., 580 Waksman, Z., 90 Xiong, G., 418, 506, 535, 576, 578 Wallen, L. J., 389 Yang, C., 569 Walter, R., 127, 132 Yang, D., 389, 495–497, 500, 504–506, 540, 558, Wan, X., 586 568, 575–577, 591, 592, 595, 600, 601, 603, Wang, G., 583, 609 605, 607, 610–613 Wang, T., 610, 611, 622 Yano, K., 310 Wang, Wei, 559, 587 Yaskin, V., xvi, 473, 576, 582, 583, 586 Wang, Weidong, 516, 559, 579, 585, 586 Yaskina, M., 473, 582, 586 Wang, X.–J., 228, 462, 478, 500, 506, 558 Yau, S.–T., 462 Wang, Y., 399 Ye, D., 556, 559 Wannerer, T., 228, 361, 365, 367, 368, 619 Yost, D., 169 Weberndorfer, M., 543, 601, 608, 609 Yu, W., 577, 580, 587, 593

728 Author index

Yuan, J., 514, 516, 579, 582, 583, 585, 595, Zhong, X., 392 606 Zhou, F., 478 Yuan, S., 514, 582, 583 Zhou, J., 418, 559, 569 Zahle,¨ M., 127, 130, 132, 217, 226, 227, 235, 236, Zhu, B., 559 261, 264 Zhu, G., 569 Zaguskin, V. L., 594 Zhu, H., 583 Zaj´ıcek,ˇ L., 31, 89, 103 Zhu, M., 506 Zalgaller, V. A., 10, 15, 91, 103, 379, 390, 398 Zhu, X., 399, 559, 583, 586 Zamﬁrescu, T., xvi, 133, 134, 137, 152, 153 Zhu, X.–Y., 583 Zarichnyi, M. M., 70 Zhuang, X., 151 Zhang, G., 188, 289, 294, 389, 418, 471, 477, Ziegel, J., 321 495–497, 500, 504–506, 510, 534, 535, 540, Ziegler, G. M., xvi, 8, 112 558, 567, 568, 573–577, 582, 584–586, 591, Zinn, J., 154 ˇ 592, 595, 600, 601, 603, 605, 607, 610–613 Zivaljevic,´ R. T., 73 Zhao, C.–J., 393, 512, 516, 559, 578, 579, 583 Zong, C., xvi Zhao, G., 543 Zvavitch, A., 204, 511, 543, 565, 580, 583, 585 Zhao, L., 516 Zymonopoulou, M., 582, 583, 586

Subject index

a-type, 109 barycentric coordinates, 6 abstract duality, 43 Binet ellipsoid, 590 additive, 172, 329 Blaschke affine addition, 459, 463 combination, xix body, 463 energy, 610 diagram, 387 function, 20 –Groemer inequality, 539 hull, xix –Minkowski homomorphism, 460 isoperimetric inequality, 545 –Santalo´ inequality, 548, 563 normal vector, 544 Lp, 568 perimeter, 545 selection theorem, 63 Polya–Szeg´ o¨ principle, 613 sum, 459 quermassintegral, 515 Blaschke’s rolling theorem, 164 Sobolev inequality, 610 body of constant width, 140 surface area, 544, 545 Bonnesen inequality, 388 affinely independent, xix brightness function, 529 Aleksandrov body, 411 Brunn–Minkowski theorem, 369 Aleksandrov–Fenchel inequality, 393 Brunn–Minkowski theory Aleksandrov–Fenchel–Jessen theorem, Lp, 489 448 dual, 507 Aleksandrov’s projection theorem, 449 Busemann intersection inequality, 580 Alesker’s irreducibility theorem, 365 Busemann–Petty problem, 583 analogous (polytopes), 110 Busemann random simplex inequality, 539 antisummand, 139, 170 area centroid, 314 canal class, 440 area measure, 214, 215 cancellation law, 48, 139 Asplund sum, 517 cap, 17, 91 associated convex function, 51 body, 87 asymmetric Lp affine energy, 613 covering theorem, 91 asymmetric Lp zonotope, 609 Caratheodory’s´ theorem, 3 asymmetry class, 170, 472 category asymmetry function, 170 first, 132 Cauchy’s surface area formula, 301 Baire space, 132 centred support function, 262 ball centroaffine surface area, 555 closed, xxi centroid, 314 open, xxi centroid body Ball–Barthe inequality, 601 p-, 567 Banach–Mazur distance, 589 Orlicz, 568 barycentre function, 19 characteristic function, xx, 329

730 Subject index

Chebyshev set, 10 dilatate, xxi Choquet simplex, 532 dilatation, xxi chord power integral, 534 dimension, 7 Christoffel’s problem, 470, 472 direct sum, xx circumball, 142 direct summand, 156 circumradius, 142 directly indecomposable, 156 class reduction, 576 discriminant closed mixed, 322 ball, xxi dissection, 346 convex function, 36 distance, xxi convex hull, 6 dual halfspace, xx Lp John ellipsoid, 593 segment, xix affine quermassintegral, 515 combination Brunn–Minkowski theory, 507 affine, xix cone, 35 convex, 1 harmonic quermassintegral, 514 linear, xix mixed volume, 508 positive, 2 quermassintegral, 508 common Dupin indicatrix, 126 antisummand, 467 summand, 467 economic cap covering, 104 support plane, 103 edge, 105 concave effective domain, 20 α-, 520 ellipsoid function, 20 Lp John, 592 cone-volume measure, 501 dual, 593 cone-volume probability measure, 495 Binet, 590 conjugate face, 75 John, 587 conjugate function, 37 Legendre, 590 constant width, 140 Loewner, 587 continuous convex set, 59 LYZ-, 591 convex, 1 elliptic type, 547 body, 8 p, 555 combination, 1 epigraph, 20 cone, 1 equidissectable, 346 floating body, 561 Euler function, 19 characteristic, 214, 230 hull, 2 point, 127 convolution body, 381, 561 relation, 337 covariogram, 528 -type relation, 339 Crofton’s intersection formula, 245 Euler’s theorem, 115 cross covariogram, 530 even measure, 192 cross measure, 607 even valuation, 353 current representation, 217 exceptional, 103 curvature exposed centroid, 314 r-, 76 function, 545 normal vector, 85 image, 547 support plane, 85 measure, 214 r-skeleton, 76 cyclically monotonic, 30 face, 75 cylinder, 341 normal vector, 85 point, 18 diameter, xxi, 49 support plane, 85 difference body, 140, 529 exterior normal vector, 11, 81 inequality, 530 external angle, 109

Subject index 731

extreme, 85 generating measure, 193 (K1,...,Kn−1), 87 generating set, 160, 167 r-, 76 generic, 132 normal vector, 85 Geometric Barthe inequality, 597 support plane, 85 Geometric Brascamp–Lieb inequality, 597 normal vector, 85 geometric convex function, 41 support plane, 85 geometric Minkowski combination, 494 point, 16 geominimal surface area, 549 ray, 17 gradient, 26 Grassmannian, xxii face, 16 Grothendieck group, 172 i-, 74 -function, 78 half-flat, xx facet, 74, 105 half-open segment, xx Fenchel conjugate, 40 Hanner polytope, 564 first category, 132 harmonic first fundamental form, 114 p-combination, 512 flag measure, 227 Blaschke addition, 513 flat, xx Blaschke linear combination, 513 floating body, 561 quermassintegral, 514 flotation body, 560 Hausdorff form body, 386 asymmetry index, 390 formula distance, 61 Cauchy’s surface area, 301 measure, xxi Crofton’s intersection, 245 metric, 61, 68 Kubota’s, 301 hedgehog, 526 principal kinematic, 239 Helly’s theorem, 4 Steiner, 208 herisson,´ 526 local, 213 Hessian matrix, 28 fully additive, 330 Hessian measure, 228 function homogeneous, 341 affine, 20 homothet, xxi barycentre, 19 homothetic, xxi characteristic, xx positively, xxi concave, 20 homothety, xxi conjugate, 37 hull convex, 19 affine, xix gauge, 53 closed convex, 6 indicator, xx, 21 convex, 2 support, 44 linear, xix positive, 2 gauge function, 53 hyperbolic polynomial, 323 gauge transform, 42, 55 hyperbolicity cone, 323 Gauss map, 113 hyperplane, xx Gauss–Kronecker curvature, 115 hyperspace, 70 general Brunn–Minkowski theorem, 406 improper rigid motion, xxii general relative position, 474 inclusion measure, 418 generalized inclusion–exclusion principle, 330 M-body, 191 indecomposable, 164 curvature measure, 213 indecomposable pair, 171 principal curvature, 129 independent triangle body, 201 affinely, xix zonoid, 195 index function, 231 generated Minkowski class, 190 indicator function, xx, 21

732 Subject index

inequality John position, 596 Lp Busemann–Petty centroid, 568 John’s theorem, 587 affine Lp Sobolev, 610, 611 affine Sobolev, 610 Kirchberger’s theorem, 14 Aleksandrov–Fenchel, 393 Klain Ball–Barthe, 601 embedding, 356 Blaschke–Groemer, 539 function, 356 Blaschke–Santalo,´ 548, 563 map, 356 Bonnesen, 388 Kneser–Suss¨ inequality, 460 Busemann intersection, 580 Krein–Milman theorem, 18 Busemann random simplex, 539 Kubota’s integral recursion, 301 difference body, 530 ffi Geometric Barthe, 597 Lφ a ne surface area, 556 ffi Geometric Brascamp–Lieb, 597 Lψ a ne surface area, 557 n geominimal surface area, 549 L -star, 511 Holder’s¨ for integrals, 509 Lp isodiametric, 382 addition, 490 isoperimetric, 376, 382 Brunn–Minkowski theory, 489 Jensen, 20 John ellipsoid, 592 Kneser–Suss,¨ 460 Minkowski problem, 498 Minkowski’s, 382 Minkowski valuation, 617 first, 382 dual affine surface area, 559 for integrals, 509 dual geominimal surface area, 559 second, 382 harmonic Blaschke addition, 513 Petty projection, 572 metric, 71 affine, 579 surface area measure, 494 Prekopa–Leindler,´ 373 zonoid, 606 reverse Brunn–Minkowski, 380 -norm, 602 Rogers–Shephard, 530 lattice endomorphism, 34 Urysohn, 382 Legendre Zhang projection, 573 ellipsoid, 590 infimal convolution, 39 transform, 40 inner radius, 143 transformation, 40 inradius length measure, 465 relative, 148 Levy–Prokhorov´ distance, 480 internal, 7 lift zonoid, 205 intersection body, 580 line-free, 16 Lp, 581 lineality space, 16 mixed, 583 linear intersectional family, 330 combination, xix intrinsic (r + 1)-moment, 314 hull, xix intrinsic volume, 208, 214 parameter system, 542 irreducible, 169 linearity direction, 25 isodiametric inequality, 382 local parallel set, 209 isoperimetric inequality, 376, 382 local parallel volume, 121 affine, 545 local Steiner formula, 213 isotropic locally determined, 215 constant, 605 locally embeddable, 157 measure, 595 locally similar (polytopes), 110 normalized, 596 Loewner ellipsoid, 587 position, 604 Loewner position, 596 log-concave function, 517 Jensen’s inequality, 20 lower curvature, 113 Jessen radius of curvature, 125 lower radius of curvature, 113 John ellipsoid, 587 lower semi-continuous hull, 38

Subject index 733

LYZ body, 611 width integral, 513 LYZ ellipsoid, 591 moment body, 566 asymmetric Lp, 568 M-addition, 495 moment matrix, 589 maximal common summand, 171 moment vector, 312 McMullen decomposition, 341 multiplier transformation, 180 meagre, 132 mean nearest-point map, 9 curvature, 115 normal, 125 dual affine quermassintegral, 516 bundle, 127, 132, 209 section body, 265, 477 cone, 81 width, 50, 173, 297, 519 point, 127 measurable, 209 vector, xx, 106 measure normalized convex body, 164 Hausdorff, xxi normalized isotropic measure, 596 of non-convexity, 144 metric entropy, 72 o-symmetric, xx metric projection, 9 obverse, 42 metric tensor, 316 odd valuation, 353 minimal common anti-summand, 170 open ball, xxi minimal pair, 170, 472 order cancellation law, 139 Minkowski order isomorphism, 34 addition, 48 Orlicz additive, 48, 172 centroid body, 568 class, 189 projection body, 575 difference, 146 sum, 496 endomorphism, 177 zonotope, 609 functional, 53, 208 outer linear, 172 normal vector, 11, 81 problem parallel body, 128, 208 Lp, 498 unit normal vector, 106 subtraction, 146 volume ratio, 599 symmetrization, 181 tensor, 317 p- valuation, 364, 586, 616 addition, 490 ffi Lp, 617 a ne surface area, 554 Minkowski’s curvature function, 554 existence theorem, 455 geominimal surface area, 555 inequality, 382 linear combination, 490 theorem, 17 scalar multiplication, 490 Minkowskian integral formulae, 297, 309 sum, 490 mixed tangential body, 86 p-affine surface area, 559 parallel body affine surface area, 548 inner, 148 area measure, 279, 280 outer, 148 body, 262, 463 parallel chord movement, 543 discriminant, 124, 322 perfect face, 84 integral, 521 Petty projection inequality, 572 intersection body, 583 affine, 579 moment tensor, 321 polar moment vector, 313 Lp addition, 512 projection body, 570 body, 32 quermassintegral, 427 curvature image, 546 valuation, 345 projection body, 570 volume, 277, 280 set, 32

734 Subject index

polarization, 325 Radon’s theorem, 3 polarization formula, 277 rational homogeneous, 341 polynomial ray, xx hyperbolic, 323 recession cone, 16 valuation, 363 reduced pair, 170, 472 polytope, 3, 104 reducible, 169 porous, 136 regular, 83 position, 596 normal vector, 87 positive supporting halfspace, 84 basis, 15 relative combination, 2 boundary, xx hull, 2 indecomposability, 171 reach, 224 interior, xx positively homogeneous, 24 residual, 132 positively homothetic, xxi reverse Prekopa–Leindler´ inequality, 373 Brunn–Minkowski inequality, 380 principal curvature, 114, 127 isoperimetric inequality, 599 generalized, 129 second fundamental form, 116 principal kinematic formula, 239 spherical image, 88 principal radius of curvature, 116 map, 88 projection body, 302, 569 Weingarten map, 116 Lp, 575 rigid motion, xxi asymmetric, 575 Rogers–Shephard inequality, 530 polar, 575 rolling theorem, 164, 168 L p mixed, 587 roots of Steiner polynomial, 224 Orlicz, 575 rotation, xxii projection generating measure, 308 rotor, 189 proper (convex function), 19 Santalo´ point, 546 (face), 74 Santalo´ region, 562 (rigid motion), xxii second fundamental form, 114 separation, 12 semiaxis function, 59 separate, 12 quasi-concave, 520 shadow system, 542 quermassintegral, 208, 213 shaking, 539 quermassvector, 314 Shapley–Folkman lemma, 141 r-singular, 83 Shapley–Folkman–Starr theorem, 143 r-skeleton, 76 shearing, 347 radial similarity, xxii p-combination, 508 simple pth mean body, 574 (polytope), 109 addition, 507 (valuation), 346 Blaschke linear combination, 509 simplex, 3 Blaschke–Minkowski homomorphism, simplicial, 109 512 singular relative position, 251 Blaschke sum, 509 slicing problem, 606 function, 57 sliding freely, 156 linear combination, 507 smooth, 83 map, 89 smooth valuation, 366 metric, 507 Sobolev distance, 71 sum, 507 special position, 238, 240 valuation, 620 speciﬁc curvature, 274

Subject index 735

spherical telescoping, 443 harmonic, 623 theorem image, 88 Aleksandrov’s projection, 449 map, 88, 113 Aleksandrov–Fenchel–Jessen, 448 Laplace operator, 119 Alesker’s irreducibility, 365 stable convex body, 78 Blaschke selection, 63 star body, 57, 507 Brunn–Minkowski, 369 star duality, 507 general, 406 starshaped, 57 Caratheodory,´ 3 Steiner Euler, 115 ball, 421 Groemer’s extension, 338 formula, 208 Helly, 4 point, 50, 315, 319 John, 587 abstract, 181 Kirchberger, 14 polynomial, 208 Krein–Milman, 18 roots, 224 Minkowski, 17 symmetral, 536 Minkowski’s existence, 455 symmetrization, 536 Radon, 3 Steinitz’s theorem, 14 coloured, 5 Straszewicz’s theorem, 18 Shapley–Folkman–Starr, 143 strict separation, 12 Steinitz, 14 strict subspace concentration inequality, Straszewicz, 18 504 Tverberg, 8 strictly convex, 87 thickness, 49 strong separation, 12 touching cone, 85 strongly isomorphic, 109 translate, xxi subdiﬀerential, 29 subgradient, 29 translation, xxi sublevel set, 20 invariant, 341 sublinear, 24 mixture, 262 subspace concentration condition, 504 triangle body, 201 summand, 139, 147, 156 type sup-convolution, 517 a-, 109 support, 11 typical, 132 cone, 81 element, 59, 127, 209 umbilical point, 484 function, 44, 518 unit ball, xxi measure, 213 unit sphere, xxi number, 276 universal approximating class, 186 plane, xx, 11, 45 universal convex body, 200 set, 45 upper vector, 293 curvature, 113 supporting halfspace, 11, 45 level set, 520 surface area measure, 214, 215 radius of curvature, 113 Lp, 494 Urysohn’s inequality, 382 surface body, 562 symmetric diﬀerence metric, 71 valuation, 172, 329 vertex, 105 T-equidissectable, 346 vertical (halfspace), 36 tangent space, 112 virtual polytope, 526 tangential body, 86 volume functional, 57 tangential radius of curvature, 126 volume product, 563

736 Subject index

volume ratio, 597 integral, 514 inner, 597 Wills functional, 310 outer, 599 Wulﬀ shape, 411, 418

weakly Zhang projection inequality, 573 continuous, 348 zonal signed measure, 177 monotonic, 179 zonoid, 191, 192 positive measure, 179 Lp, 606 weighted ﬂoating body, 562 equation, 203 Weingarten map, 113 generalized, 195 width, 49 zonotope, 191 constant, 140 antisymmetric Lp, 609 function, 49, 140 Orlicz, 609