Stability results for convex bodies in geometric tomography

Abstract

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.

Valuations and surface area measures

“…Note that pth powers of support functions are homogeneous of degree p. Moreover, Kiderlen [19] showed that differences of pth powers of support functions are dense in the space of continuous functions on S n−1 . So the last equation immediately implies that L p surface area measures are GL(n) contravariant of degree p.…”

A Fourier transform approach to Christoffel’s problem

“…This observation, which was kindly pointed out to us by Markus Kiderlen, allows us to obtain a strengthened version of one of his results [23].…”

“…We see that I 1 converges uniformly to the integral term in (23). Our goal is to show that I 2 and I 3 approach zero uniformly and also that the limit of…”

“…Using a differential equation for g n , established by Berg, the same ideas as in Step 2 allow us to compute the second derivative and prove (23).…”

“…If we further assume that f satisfies (11), then h = f , and so f is a difference of first surface area measures. It follows that the index 2 [(n + 1)/4] in [23,Proposition 3.2] can be replaced by 1.…”