Abstract
This survey is devoted to papers in the area of integral geometry published during the last ten years.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
S. A. Demidova, “Application of the theory of integral invariants of Lie groups to certain aspects of integral geometry,” Dissertation, Kharkov (1956).
G. I. Drinfel’d, “On measure in Lie groups,” Khar’kov. Gos. Univ. Uch. Zap., 29 (1949)
G. I. Drinfel’d, “Some basic formulas of integral geometry. II,” Khar’kov. Gos. Univ. Univ. Uch. Zap., 40 (1952)
G. I. Drinfel’d, “Theory of integral invariants and integral geometry,” Proc. Third Ail-Union Math. Congress, Vol. I [in Russian], Akad. Nauk SSSR, Moscow (1956), p. 151.
G. I. Drinfel’d, “Computation of an affine arc and of an affine area,” Ukr. Geometr. Sb., (1968).
G. I. Drinfel’d and Kyong Kim, “Integral invariants of completely integrable Pfaffian systems of differential equations,” Dopovidi Akad. Nauk Ukrain.RSR, No. 6, 713–716 (1963).
G. I. Drinfel’d and A. V. Lutsenko, “The measure of the set of second-order curves,” Dopovidi Akad. Nauk Ukrain.RSR, No. 1, 14–17 (1965).
Kyong Kim, “Integral invariants of a completely integrable system of equations in terms of total differentials,” Vestnik Khar’kov. Gos. Univ., Ser. Mekh.-Mat., No. 3, 51–64 (1965).
A. V. Lutsenko, “The measure of sets of geometric elements and of their subsets,” Ukr. Geometr. Sb., No. 1, 39–57 (1965).
Yu. G. Reshetnyak, “Integral-geometric methods in the theory of curves,” Proc. Third AU-Union Math. Congress, Vol. I [in Russian], Akad. Nauk SSSR, Moscow (1956), p. 164.
L. Santaló, Introduction to Integral Geometry [Russian translation], Izd. Inostr. Lit., Moscow (1956), 183 pp.
M. Stoka, “The measure of families of manifolds in the space E3,” Rev. Math. Pures et Appl. (RPR), 4(2):305–316 (1959).
E. G. Ushakova, “Conditions for the existence of an integral invariant of a transitive group,” Khar’kov. Gos. Univ. Uch. Zap., 115 (1961)
N. G. Chebotarev, “The determination of volume in Lie groups,” Zap. Mat. Otd. Fiz.-Mat. Fak. i Khar’kov. Mat. Obshch., 14:3–20 (1937),
N. G. Cebotarev, Lie Groups [in Russian], Moscow-Leningrad (1940).
L. M. Yurtova and A. V. Lutsenko, “Measures of sets of pairs,” Ukr. Geometr. Sb., (1968).
I. M. Yaglom, “Integral geometry in the set of line elements,” Appendix to the book: L. Santaló, Introduction to Integral Geometry [Russian translation], Izd. Inostr. Lit., Moscow (1956), 183 pp.
R. E. Alemany, Valores numericos de ciertas constantes relacionadas con volumenes mixtos de cuerpos convexos. Rev. Union mat. argent. y Asoc. fis. argent., 21(3):113–118 (1963).
A. S. Besicovitch, “On the set of directions of linear segments on a convex surface,” in: Convexity, Amer. Math. Soc. (1963), pp. 24–25.
W. Blaschke, Vorlesungen über Integralgeometrie, 3 ed., VEB Dtsch. Verl. Wiss. (1955), Vol. 8, 130 pp.
G. Bouligand, Sur les transformations ponctuelles conservant lesaires. Gaz. mat., 14(56):1–4 (1953).
J. E. Brothers, “Integral geometry in homogeneous spaces,” Doctoral dissertation, Brown Univ. (1964), 87 pp.; Dissert. Abstr., 25(8):4717–4718 (1965).
J. E. Brothers, Integral geometry in homogeneous spaces. Trans. Amer. Math. Soc., 124(3):480–517 (1966).
H. Busemann, Areas in affine spaces. III. The integral geometry of affine area. Rend. Circolo mat. Palermo, 9(2):226–240 (1960).
G. Chakerian, Integral geometry in the Minkowski plane. Duke math. J., 29(3): 375–381 (1962).
Shiing-shenChern, On integral geometry in the Klein spaces. Ann. math., 43: 178–189 (1942).
Shiing-shen Chern, Differential Geometry and Integral Geometry, Proc. Internat. Congr. Mathematicians, 1958. Cambridge Univ. Press, Mass. (1960), pp.440–449.
Shiing-shen Chern, On the kinematic formula in integral geometry. J. Math. and Mech., 16(1):101–118 (1966).
V. Cruceanu, Asupra transformårilor infinitezimale ale unui spatiu Riemann cu pstrarea volumului. Studii i cercetri tiin. Acad. RPR. Fil. Iai Mat., 9(2): 181–188 (1958).
R. Deltheil, Probabilités géometriques, Paris, (1926), Traité du calcul des probabil. et de ses applic, tome II, fasc. II.
I. Fáry, Functionals related to mixed volumes. Illinois J. Math., 5(3):425–430 (1961).
H. Federer, Curvature measures. Trans. Amer. Math. Soc., 93(5):418–491 (1959).
W. J. Firey, Generalized convex bodies of revolution.Canad.J. Math., 19(5): 972–996 (1967).
F. Gaeta, Algunas observaciones sobre la realiracion efectiva del programa de Chern geometria integral. Math. notae., 17(1–2):1–29 (1959).
F. Gaeta, Sobre un proceso de linearizacion aplicable en problemas de geometria integral. Publs Fac. cienc, fisicomat. Univ. nac. La Plata, No. 224, pp. 17–32 (1960).
F. Gaeta, Sobre la subordinacion de la geometria integral a la teoria de la representacion de grupos mediante transformaciones lineales. Contribs. cient. Univ. Buenos Aires. Fac. cienc. exactas y natur. Mat., 2(2):27–87 (1960).
H. Giger, Ermittlung der mittleren Masszahlen von Partikeln eines Körpersystems durch Messungen auf dem Rand eines Schnittbereichs. Z. angew. Math. und Phys., 18(6):883–888 (1967).
H. Hadwiger, Über additive Funktionale k-dimensionaler Eipolyeder. Publs. mathematical, 3(1–2):87–94 (1953).
H. Hadwiger, Zur kinematischen Hauptformel der integralgeometrie. Proc. Internat. Congr. Math., Amsterdam, 2:225–226 (1954).
H. Hadwiger, Altes und Neues über konvexe Körper, Birkhäuser, Basel (1955), 116 pp.
H. Hadwiger, Minkowskis Ungleichungen und nichtconvexe Rotationskörper. Math. Nachr., 14(4–6):377–383 (1955).
H. Hadwiger, Körper im euklidischen Raum und ihre topologischen und metrischen Eigenschaften. Math. Z., 71(2):124–140 (1959).
H. Hadwiger, Über konkave und konvexe Eikörperscharen. Publs. math. Debrecan, Nos. 1–2, pp. 97–101 (1957).
H. Hadwiger, Zur Axiomatik der innermathematischen Wahrscheinlichkeitstheorie. Mitt. Verein. Schweiz. Versicherungsmathematiker, 58(2):151–165 (1958).
H. Hadwiger, Geometrische Wahrscheinlichkeiten bei Durchstichen von Geraden durch Kugelflächen. Mitt. Verein. Schweiz. Versicherungsmathematiker, 68(1): 27–35 (1968).
R. Hermann, Remarks on the foundations of integral geometry. Rend. Circolo mat. Palermo, 2(9):91–96 (1960).
Chuan-Chih Hsiung and J. K. Shahin, Affine differential geometry of closed hypersurfaces. Proc. London Math. Soc., 17(4):715–735 (1967).
K. Legrady, Symplektische Integralgeometrie. Ann. mat. pura ed appl., 41:139–159 (1956).
R. E. Luccioni, Sobre la existencia de medida para hipercuadricas singulares en espacios projéctivos. Rev. Univ. nac. Tucumán, A14(1–2):269–276 (1962).
R. E. Luccioni, Geometria integral en espacios projéctivos. Rev. Univ. nac. Tucumán, A15(1–2):53–80 (1964).
R. E. Luccioni, Sobre la existencia de medida para conjuntos de subespacios lineales en espacios afines. Rev. Univ. nac. Tucumán, A16(1–2):219–227 (1966).
G. Lüko, On the mean length of the chords of a closed curve. Israel J. Math., 4 (1):23–32 (1966).
C. D. Maraval, El problema de la aguja de Buffon en el espacio de n dimensiones. Gac. mat., 11(3–4):74–75 (1959).
G. Masotti Biggiogero, La geometria integrale. Rend. Seminar mat. e fis. Milano, 1953–1954; 25:164–231 (1955).
G. Masotti Biggiogero, Sulla geometria integrale: generalizzazione di formule di Crofton, Lebesgue e Santalo. Rev. Union mat. argent. y Asoc fis. argent., 17:125–134 (1955).
G. Masotti Biggiogero, Sulla geometria integrale: nuove formule relative agli ovaloidi. Scritti mat. onore Filippo Sibirani. Bologna, (1957), pp. 173–179.
G. Masotti Biggiogero, Nuove formule di geometria integrale relative agli ovali. Ann. mat. pura ed appl., 58:85–108 (1962).
G. Masotti Biggiogero, Nuove formule di geometria integrale relative agli ovaloidi. Rend. Ist. lombardo sci. e lettere. Sci. mat., fis., chim. e geol, A69(3):666– 685 (1962).
M. Masuyama, On a fundamental formula in bulk sampling from the viewpoint of integral geometry. Repts Statist. Applic. Res., Union Japan Scientists and Engrs, 4(3):85–89 (1956).
H. R. Müller, Über Momente ersten und zweiten Grades in der Integral-geometrie. Rend. Circolo mat. Palermo, 2(1):119–139 (1953).
Z. Nádenik, O Integralni geometri. Pokroky mat., fiz.a astron., 7(2):75–79 (1962).
A. Novikoff, “Integral geometry as a tool in pattern perception,” in: H. von Foerster and G. W. Zopf, Jr. (Eds.), Principles of Self-Organization, Pergamon Press, Oxford (1962), pp. 347–368.
T. Onodera, On hypersurfaces in certain n-dimensional spaces. Tensor, 19(1): 55–63 (1968).
M. de Resmini, Un ramo relativamente nuevo della matematica: la geometria integrale. Archimede, 13(3):134–144 (1961).
L. A. Santaló, “On the kinematic formula in spaces of constant curvature,” Proc. Internat. Congr. Math., 1954, Amsterdam (1954), pp. 251–252.
L. A. Santaló, “Sur la mesure des espaces linéaires qui coupent un corps convex et problèmes qui s’y rattachent,” Colloq. quest, réalité géom., Liège, 1955, Liège (1956) pp. 177–190.
L. A. Santaló, On the mean curvatures of a flattened convex body. Istanbul univ. fen. fac. mecm., A21(3–4):189–194 (1956).
L. A. Santaló, Un nuevo invariante afin para las figures convexes del piano y del espacio. Math. notae, 16(3–4):78–97 (1958).
L. A. Santaló, Two applications of the integral geometry in affine and projective spaces. Publs math. Debrecen, 7(1–4):226–237 (1960).
L. A. Santaló, La formula de Steiner para superficies paralelas en geometria afin. Rev. Univ. nac. Tucumán, A13(1–2):194–208 (1960).
L. A. Santaló, On the measure of sets of parallel linear subspaces in affine space. Canad. J. Math., 14(2):313–319 (1962).
L. A. Santaló, Sobre la formula fundamental cinematica de la geometria integral en espacios de curvatura constant. Math. notas, 18 (1):79–94 (1962).
L. A. Santaló, “Integral geometry of the projective groups of the plane depending on more than three parameters,” An. stiint, Univ. Iasi, Sec. Ia, 11b:307–335 (1965).
L. A. Santaló, Horocycles and convex sets in hyperbolic plane. Arch. Math., 18(5):529–533 (1967).
L. A. Santaló, Grupos del piano respecto de los cuales los conjuntos de puntos y de rectas admiten una medida invariante. Rev. Union mat. argent, y Asoc. fis. argent., 23(3):119–148 (1967).
M. Stoka, Msura unei multimi de varietâi dintr-un spatiu Rn. Bu1 stint. Acad. R. P. Rumîne. Sec. mat. i fiz., 7(4):903–937 (1955).
M. Stoka, Asupra msurii mulimii cercurilor din plan. Gaz. mat. i fiz., A7(10): 556–559 (1955).
M. Stoka, Asupra subgrupurilor unui grup Gr msurabil. Comun. Acad. RPR, 6(3): 393–394 (1956).
M. Stoka, Asupra grupurilor Gr msurabile din plan. Bul. tiin. Acad. RPR. Sec. mat. i fiz., 9(2):341–380 (1957).
M. Stoka, Asupra grupurilor Gr msurabile dintr-un spatiu Rn. Comun. Acad. RPR, No. 6, pp. 581–585 (1957).
M. Stoka, Asupra msurii multimilor de varietâi dintr-un spaiu euclidian En. Comun. Acad. RPR, No. 3, pp. 313–317 (1957). ’
M. Stoka, Geometria integrale in uno spazio euclideo En. Boll. Unione mat. ital., 13(4):170–485 (1958).
M. Stoka, Invariantii integrali ad unui grup Lie de Transformri. An. Univ. “C. I. Parchon”. Ser. stiint. natur, No. 20, pp. 33–35 (1958).
M. Stoka, Asupra grupurilor Gr msurabile dintr-un spatiu En. Comun. Acad. RPR, 9(1):5–10 (1959).
M. Stoka, Géométrie intégrale dans un espace En. Rev. math. pures et appl., (RPR), 4 (1):123–156 (1959),
M. Stoka, Congruences de variétés mesurables dans un espace En. Rev. math. pures et appl. (RPR), 4(3):431–439 (1959).
M. Stoka, Famiglie di varietà misurabili in uno spazio En. Rend. Circolo mat. palermo, 8(2):192–205 (1959).
M. Stoka, Integralgeometrie in einem Riemannschen Raum Vn Rev. math. pures et appl. (RPR), 5(1):107–120 (1960).
M. Stoka, Geometrie integral intr-un spatiu riemannian Vn. Studii i cercetri mat. Acad. RPR. Fil. Cluj. 11(2):381–395 (1960).
M. Stoka, Das Mass der Untersysteme von Mannigfaltigkeiten in einem Raum Xn. Rev. math. pures et appl. (RPR), 5(2):275–286 (1960).
M. Stoka, Geometrie integral într-un spatiu euklidian En, Lucrrile conf. geometrie i topol., 1958, Bucuresti, Acad. RPR (1962) pp. 109–116.
M. Stoka, Familii de varietti msurabile intr-un spatiu riemannian V3 cu curbura constant negativ. Studii i cercetari mat. Acad. RPR, 14(3):365–376 (1963).
M. Stoka, Geometria integral, Bucuiesti, Acad. RPR (1967), 238 pp.
R. Sulanke, Die Verteilung der Sohnenlängen an ebenen und räumlichen Figuren. Math. Nachr., 23 (1):51–74 (1961).
R. Sulanke, Integralgeometrie ebener Kurvennetze. Acta math. Acad. scient. hung., 17(3–4):233–261 (1966).
R. Sulanke, Croftonsche Formeln in Kleinschen Räumen. Math. Nachr., 32(3–4): 217–241 (1966).
K. Tekse, Riemann-térz intégralgeometriá jának néhény problémájáról. Magyar tud. acad. Mat. és. fiz. tud. aszt. közl., 11(3):289–304 (1961).
R. Trandafir, Problems of integral geometry of lattices in an Euclidean space E3. Boll. Unione math. ital., 22(2):228–235 (1967),
R. Trandafir, Problems of integral geometry of lattices in a riemannian space V2 with constant curvature. Boll. Unione math. ital., 1(2):244–248 (1968).
S. Ueno, On the densities in a two-dimensional generalized space. Mem. Fac. Sci. Kyûsyû Univ., A9 (1):65–77 (1955).
A. E. Vidal, A generalization of integral invariants. Proc. Amer. Math. Soc., 10(5):721–727 (1959).
A. E. Vidal, Generalización de los invariantes integrales y aplicación a la geometria integral en los espacios de Klein y de Riemann. Collect, math., 12(2): 71–102 (1960).
G. Vrnceanu, The measurability of Lie groups. Ann. polon. math., 15(2):179–188 (1964).
S. Watanabe, On hypersurfaces of spaces belonging to certain transformation group. Acta math. Acad. scient. hung., 17(1–2):137–145 (1966).
S. Watanabe, On hypersurfaces of spaces belonging to a certain transformation group. II. Tensor, 18 (1):90–96 (1967).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1972 Plenum Press, New York
About this chapter
Cite this chapter
Drinfel’d, G.I. (1972). Integral Geometry. In: Gamkrelidze, R.V. (eds) Algebra and Geometry. Progress in Mathematics, vol 12. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0507-2_4
Download citation
DOI: https://doi.org/10.1007/978-1-4757-0507-2_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-0509-6
Online ISBN: 978-1-4757-0507-2
eBook Packages: Springer Book Archive