zbMATH — the first resource for mathematics

On uniqueness of reconstruction of the form of convex and visible bodies from their projections. (English) Zbl 0841.52001
Bokut’, L. A. (ed.) et al., Third Siberian school on algebra and analysis. Proceedings of the third Siberian school, Irkutsk State University, Irkutsk, Russia, August 30-September 4, 1989. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 163, 35-45 (1995).
The author generalizes the following result of W. Süss [Tôhoku Math. J. 35, 47-50 (1932; Zbl 0003.41001)]. Let $$K$$ be a convex body in $$\mathbb{R}^n$$ and denote by $$(K |E_i)$$ the orthogonal projection of $$K$$ onto an $$i$$-dimensional plane $$E_i$$. Süss showed that if for two convex bodies $$K$$ and $$L$$ the projection $$(K |E_{n -1})$$ onto any hyperplane $$E_{n-1}$$ is a translate of $$(L |E_{n-1})$$ then $$K$$ and $$L$$ themselves are translates of each other. First the author shows that Süss’ condition that $$(K |E_{n - 1})$$ is a translate of $$(L|E_{n -1})$$ need not be assumed for all $$E_{n-1}$$ to get the same conclusion. Next he derives the following result: If for convex bodies $$K$$ and $$L$$ all projections $$(K|E_2)$$ and $$(L|E_2)$$ onto 2-dimensional planes are congruent and have no rotation symmetries then $$K$$ and $$L$$ are either translates of each other or they are centrally symmetric to each other (i.e. $$L = p - K$$ for a suitable point $$p$$). If in the assumption “congruent” is replaced by “similar” then “centrally symmetric” is to be replaced by “homothetic”. The author can also weaken the assumption “no rotation symmetries”, or the assumption “convex body”, and still derives similar results. The main step used in the proofs is to prove the related result for $$n = 3$$.
For the entire collection see [Zbl 0816.00016].
Reviewer: B.Kind (Bochum)
MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 53C65 Integral geometry 52A15 Convex sets in $$3$$ dimensions (including convex surfaces)