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 Mat. Sb. (N.S.), 1981, Volume 116(158), Number 4(12), Pages 539–546 (Mi msb2482)

Uniqueness and stability of the solution of a problem of geometry in the large

Yu. E. Anikonov, V. N. Stepanov

Abstract: This paper considers the problem of determining a convex surface from the area $F(n)$ of its orthogonal projection on any plane $(x,n)=0$ and the area $S(n)$ of the portion of the surface illuminated in the direction $n$. It is proved that in a certain class a convex surface is uniquely defined (up to translation) by a function $\varphi(n)=2aF(n)+bS(n)$ for $a\ne0$, $b\ne0$, $a+b\ne0$. Moreover, the surface is analytic if and only if $\varphi(n)$ is an analytic function on the unit sphere. The surface is shown to be stable, and a quantitative estimate related to stability is given.
Bibliography: 6 titles.

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English version:
Mathematics of the USSR-Sbornik, 1983, 44:4, 483–490

Bibliographic databases:

UDC: 514.17
MSC: 53C45

Citation: Yu. E. Anikonov, V. N. Stepanov, “Uniqueness and stability of the solution of a problem of geometry in the large”, Mat. Sb. (N.S.), 116(158):4(12) (1981), 539–546; Math. USSR-Sb., 44:4 (1983), 483–490

Citation in format AMSBIB
\Bibitem{AniSte81} \by Yu.~E.~Anikonov, V.~N.~Stepanov \paper Uniqueness and stability of the solution of a~problem of geometry in the large \jour Mat. Sb. (N.S.) \yr 1981 \vol 116(158) \issue 4(12) \pages 539--546 \mathnet{http://mi.mathnet.ru/msb2482} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=665854} \zmath{https://zbmath.org/?q=an:0507.53041|0481.53050} \transl \jour Math. USSR-Sb. \yr 1983 \vol 44 \issue 4 \pages 483--490 \crossref{https://doi.org/10.1070/SM1983v044n04ABEH000980} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Stefano Campi, “Recovering a centred convex body from the areas of its shadows: a stability estimate”, Annali di Matematica, 151:1 (1988), 289
2. Golubyatnikov V., “Stability Problems of the Reconstruction of Some Compacts From their Projections”, 322, no. 1, 1992, 20–21
3. V. N. Stepanov, “Opredelenie kompaktnykh mnozhestv po funktsionalam ot nikh”, Sib. elektron. matem. izv., 2 (2005), 167–185
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