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Markus Kiderlen

Stability results for convex bodies in geometric tomography

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  • Department of Mathematical Sciences

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>2, 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 Holder 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.

Original languageEnglish
JournalIndiana University Mathematics Journal
Volume57
Pages (from-to)1999-2038
Number of pages40
ISSN0022-2518
DOIs
Publication statusPublished - 2008

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