Uniqueness and Stability of the Solution of a Problem of Geometry in the Large

Anikonov, Ju E; Stepanov, V. N.

doi:10.1070/sm1983v044n04abeh000980

Cited by 3 publications

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As the integral transforms involved here are the cosine transform and the hemispherical transform, Theorem 4 implies a stability inequality with Hölder exponent arbitrarily close to 2/(n(n + 1)). A stability result for this data in the case n = 3 was also derived in [2], but requires that a 6th-order derivative of F (K, ·) is uniformly small.…”

Section: Hug and Schneidermentioning

confidence: 99%

Stability results for convex bodies in geometric tomography

Kiderlen¹

2008

Indiana Univ. Math. J.

View full text Add to dashboard Cite

Abstract. We consider the question in how far a convex body (non-empty compact convex set) K in n-dimensional space is determined by tomographic measurements (in a broad sense). Usually these measurements are derived from K by geometrical operations like sections, projections and certain averages of those. We restrict to tomographic measurements F (K, ·) that can be written as function on the unit sphere and depend additively on an analytical representation Q(K, ·) of K. The first main result states that F (K, ·) is a multiplier-rotation operator of Q(K, ·) whenever the tomographic data depends continuously and rotationally covariant on K. For n ≥ 3, these operators are classical multiplier transforms.We then turn to stability results stating that two convex bodies whose tomographic measurements are close to one another must be close in an appropriate metric on the family of convex bodies. We improve the Hölder exponents of known stability results for these transforms. The key idea for this improvement is to use the fact that support functions of convex bodies are elements of any spherical Sobolev space of derivative order less than 3/2. As the analytical representation Q(K, ·) may be a power of the support function, a power of the radial function, or a surface area measure, the class of tomographic data considered here is quite large. This is illustrated by many examples ranging from classical projection and section functions to directed tomographic transforms.

show abstract

Section: Hug and Schneidermentioning

confidence: 99%

Stability results for convex bodies in geometric tomography

Kiderlen¹

2008

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…( )jku( )k L1(S n¡ 1 ) µ c(n)k n=2¡1 ku( )k L1(S n¡ 1 )[17] and ¶ k = O(e ¡² k ) (by Corollary 3), the Fourier coe¯cients b function f ( ) decrease exponentially with increasing k. Hence, f ( ) is analytic on S n¡1[17] 6…”

mentioning

confidence: 99%

First-kind equations on a sphere and some problems of convex geometry

Stepanov¹

2003

Journal of Inverse and Ill-posed Problems

View full text Add to dashboard Cite

| The properties of integral operators (with a kernel depending on scalar product) acting on Banach spaces of measures and functions on the sphere S n¡ 1 are studied. A theorem of unique solvability of a¯rst-kind equation for measure is proved. Asymptotic formulae for eigenvalues of the kernel are derived. The results are used in proving theorems on the unique reconstructibility of a closed convex hypersurface from its curvature integrals and on its smoothness. Existence of a centrally symmetric closed convex hypersurface with a given curvature integral of projection is also demonstrated.State Technical University. E-Mail: stepaw49@rol.ru Brought to you by | Tokyo Daigaku Authenticated Download Date | 5/25/15 4:40 PM Brought to you by | Tokyo Daigaku Authenticated Download Date | 5/25/15 4:40 PM

show abstract

Recovering a centred convex body from the areas of its shadows: a stability estimate

Campi

1988

Annali di Matematica pura ed applicata

View full text Add to dashboard Cite

Uniqueness and Stability of the Solution of a Problem of Geometry in the Large

Cited by 3 publications

References 0 publications

Stability results for convex bodies in geometric tomography

Stability results for convex bodies in geometric tomography

First-kind equations on a sphere and some problems of convex geometry

Recovering a centred convex body from the areas of its shadows: a stability estimate

Contact Info

Product

Resources

About