Determination of compact subsets from functionals on them
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 2 (2005), pp. 167-185.

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The problems of determination of convex bodies and non-convex analytical manifolds in $R^n$ from projections, curvature integrals, sections and measures of sections of its polar are considered.
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V. N. Stepanov. Determination of compact subsets from functionals on them. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 2 (2005), pp. 167-185. http://geodesic.mathdoc.fr/item/SEMR_2005_2_a11/

[1] Aleksandrov A. D., “K teorii smeshannykh ob'emov vypuklykh tel. Rasshirenie nekotorykh ponyatii teorii vypuklykh tel”, Matem. sbornik, 2:5 (1937), 947–972 | MR | Zbl

[2] Aleksandrov A. D., “K teorii smeshannykh ob'emov vypuklykh tel. Novye neravenstva mezhdu smeshannymi ob'emami i ikh prilozheniya”, Matem. sbornik, 2:6 (1937), 1205–1238 | Zbl

[3] Aleksandrov A. D., “K teorii smeshannykh ob'emov vypuklykh tel. Rasprostranenie dvukh teorem Minkovskogo o vypuklykh mnogogrannikakh na proizvolnye vypuklye tela”, Matem. sbornik, 3:1 (1938), 27–46 | Zbl

[4] Anikonov Yu. E., “Zamechaniya o vypuklykh poverkhnostyakh”, Sibirskii matem. zhurnal, 9:6 (1968), 1413–1415 | MR | Zbl

[5] Anikonov Yu. E., Golubyatnikov V. P., “K voprosu edinstvennosti resheniya obratnykh zadach rasseyaniya”, Obratnye zadachi dlya differentsialnykh uravnenii matematicheskoi fiziki, Sb. nauchn. tr., VTs SO RAN SSSR, Novosibirsk, 1978, 13–17 | MR

[6] Anikonov Yu. E., Stepanov V. N., “Formula obrascheniya v odnoi zadache integralnoi geometrii”, DAN SSSR, 318:2 (1991), 265–266 | MR | Zbl

[7] Anikonov Yu. E., Stepanov V. N., “Edinstvennost i ustoichivost resheniya odnoi zadachi geometrii v tselom”, Matem. sbornik, 116:2 (1981), 539–546 | MR | Zbl

[8] Bonnezen T., Fenkhel V., Teoriya vypuklykh tel, Fazis, Moskva, 2002

[9] Golubyatnikov V. P., “Ob odnoznachnoi vosstanovimosti vypuklykh i obozrimykh kompaktov po ikh proektsiyam”, Matem. sbornik, 181:5 (1991), 611–621 | MR

[10] Golubyatnikov V. P., “O vosstanovlenii formy tela po ego proektsiyam”, DAN SSSR, 262:3 (1982), 521–522 | MR | Zbl

[11] Kuzminykh A. V., “O vosstanovimosti vypuklogo tela po mnozhestvu ego proektsii”, Sibirskii matem. zhurnal, 25:2 (1984), 145–150 | MR

[12] Leikhtveis K., Vypuklye mnozhestva, Nauka, Moskva, 1985 | MR

[13] Materon Zh., Sluchainye mnozhestva i integralnaya geometriya, Mir, Moskva, 1978 | MR

[14] Pogorelov A. V., Chetvertaya problema Gilberta, Nauka, Moskva, 1974 | MR

[15] Pogorelov A. V., Mnogomernaya problema Minkovskogo, Nauka, Moskva, 1975 | MR | Zbl

[16] Sobolev S. L., Vvedenie v teoriyu kubaturnykh formul, Nauka, Moskva, 1974 | MR

[17] Stepanov V. N., O vosstanovlenii nechetnoi funktsii na sfere po znacheniyam ee integralov po poverkhnostyam dolek, Dep. V VINITI 28.09.2001. No 2057-V2001, Omskii gos. tekhn. un-t., Omsk, 2001.

[18] Stepanov V. N., “Nekotorye voprosy geometrii vypuklykh poverkhnostei”, Issledovanie korrektnosti obratnykh zadach i nekotorykh operatornykh uravnenii. Sb. nauchn. tr., VTs SO RAN SSSR, Novosibirsk, 1981, 65–75

[19] Stepanov V. N., “O edinstvennosti resheniya obratnykh zadach rasseyaniya”, Differentsialnye uravneniya, 18:4 (1982), 656–663 | MR | Zbl

[20] Khelgason S., Gruppy i geometricheskii analiz, Mir, Moskva, 1987 | MR

[21] Burton G., “Sections of convex bodies”, J. London Math. Soc. (2), 12 (1976), 331–336 | DOI | MR | Zbl

[22] Bateman H., Erdelyi A., Higher Transcendental Functions, v. 2, 1955

[23] Fallert H., Goodey P., Weil W., “Spherical projections and centrally symmetric sets”, Adv. Math., 129 (1997), 301–322 | DOI | MR | Zbl

[24] Funk P., “Über eine geometrische Anwendung der Abelschen Integralgleichung”, Math. Ann., 77 (1916), 129–135 | DOI | MR

[25] Gardner R. J., Geometric tomography, Cambridge University Press, Cambridge, 1995 | MR

[26] Gardner R. J., Volcic A., “Tomography of convex and star bodies”, Adv. Math., 108 (1994), 367–399 | DOI | MR | Zbl

[27] Gardner R., Soranzo A., Volčič A., “On the determination of star and convex bodies by section function”, Discrete and Computational Geometry, 21 (1999), 69–85 | DOI | MR | Zbl

[28] Gardner R., Koldobsky A., Schlumprecht T., “An analytic solution to the Busemann–Petty problem on sections of convex bodies”, Ann. of Math. (2), 149 (1999), 691–703 | DOI | MR | Zbl

[29] Golubyatnikov V. P., Uniqueness questions in reconstruction of multidimensional objects from tomography-type projection data, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 2000

[30] Goodey P., “Radon transforms of projection functions”, Math. Proc. Camb. Phil. Soc., 123 (1998), 159–168 | DOI | MR | Zbl

[31] Goodey P., Weil W., “Centrally symmetric convex bodies and Radon transform on higher order Grassmannians”, Mathematika, 38 (1991), 117–133 | DOI | MR

[32] Goodey P., Weil W., “Centrally symmetric convex bodies and the spherical Radon transform”, J. Diff. Geom., 35 (1992), 675–688 | MR | Zbl

[33] Goodey P., Schneider R., Weil W., “On the determitation of convex bodies by projection functions”, Bull. London Math. Soc., 29 (1997), 82–88 | DOI | MR | Zbl

[34] Grinberg E., Quinto E., “Analytic continuation of convex bodies and Funk's characterization of the sphere”, Pacific J. of Math., 201 (2001), 309–322 | DOI | MR | Zbl

[35] Grinberg E., Zhang G., “Convolutios, transform and convex bodies”, Proc. London Math. Soc., 78 (1999), 77–115 | DOI | MR | Zbl

[36] Groemer H., “On a spherical integral transformation and section of star bodies”, Monatsch. Math., 126:2 (1988), 117–124 | DOI | MR

[37] Helgason S., “The totally-geodesic Radon transform on constant curvature spaces”, Contemporary Math., 113 (1990), 141–149 | MR

[38] Koldobsky A., “Inverse formula for the Blaschke–Levy representation”, Houston J. Math., 23 (1997), 95–107 | MR

[39] Koldobsky A., “An application of the Fourier transform to sections of star bodies”, Israel J. Math., 106 (1998), 157–164 | DOI | MR | Zbl

[40] Koldobsky A., Sections of star bodies and the Fourier transform, Preprint, 1991 | MR

[41] Kaasalainen M., Muinonen K., Laakso T., “Shapes and scattering properties of large irregular bodies from photometric data”, Optics Express, 8:6 (12 March, 2001), 296–301 | DOI

[42] Radon J., “Über die Bestimmung von Functionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten”, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Nat. K1, 69 (1917), 262–277 | MR | Zbl

[43] Rubin B., “Fractional integrals and wavelet transforms associated with Blaschke–Levy representations on the sphere”, Israel J. Math., 114 (1999), 1–27 | DOI | MR | Zbl

[44] Rubin B., “Inversion of fractional integrals related to the spherical Radon transform”, J. Functional Anal., 157 (1998), 470–487 | DOI | MR | Zbl

[45] Rubin B., “Spherical Radon transform and related wavelet transforms”, Appl. and Comput. Harmonic Analysis, 5 (1998), 202–215 | DOI | MR | Zbl

[46] Rubin B., “Inversion and characterization of the hemispherical transform”, J. d'Analyse Math., 77 (1999), 105–128 | DOI | MR | Zbl

[47] Schneider R., “Functions on a sphere with vanishing integrals over certain subspheres”, J. Math. Analysis and Appl., 26 (1969), 381–384 | DOI | MR | Zbl

[48] Schneider R., Convex bodies: The Brunn–Minkowski Theory, Cambridge University Press, Cambridge, 1993 | MR

[49] Ungar P., “Freak theorem about function on a sphere”, J. London Math. Soc., 29 (1954), 100–101 | DOI | MR | Zbl