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Some questions of stability in the reconstruction of plane convex bodies from projections. (English) Zbl 0597.52001
Let \({\mathcal K}\) be the class of all convex bodies in \({\mathbb{R}}^ 2\) and for H,K\(\in {\mathcal K}\) let d(H,K) and \(\theta\) (H,K) be the Hausdorff respectively Nicodym distance between H and K. The Nicodym distance is defined by \(\theta (H,K)=m(H\Delta K)\), m being the Lebesgue measure on \({\mathbb{R}}^ 2\). The convex bodies H,K\(\in {\mathcal K}\) are called \(\epsilon\)-equichordal iff for any line r, the lengths of the segments \(r\cap int H\), \(r\cap int K\) differ by less than \(\epsilon\) \((\epsilon >0).\)
The author gives the following two stability type results. 1) If H,K\(\in {\mathcal K}\) are \(\epsilon\)-equichordal then \(\theta (H,K)<C\epsilon^ 2\), where C is an absolute constant \(<14.2\). In this case the above result is an improvement of a result due to A. K. Louis and (INVALID INPUT)F. Natterer [Proc. IEEE 71, 379-389 (1983)]. 2) If H,K\(\in {\mathcal K}\) are \(\epsilon\)-equichordal and \(H\cap K\neq \emptyset\) then \(d(H,K)<3\epsilon.\)
In the last section the author studies the convexity properties of the length of the chords of \(K\in {\mathcal K}\).
Reviewer: D.Andrica

MSC:
52A10 Convex sets in \(2\) dimensions (including convex curves)
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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