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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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