Abstract
It is known that (in the sense of Baire category) most n-dimensional convex bodies are uniquely determined, up to translation or reflection, by the i-dimensional volumes of the orthogonal projections on i-planes, provided that i ∈ {2,…, n − 2}. This result is strengthened by showing that small sets of projections are sufficient for such determinations. The proof yields an extension of the result, where volumes are generalized to intrinsic volumes.
Similar content being viewed by others
References
Bauer, C, Intermediate surface area measures and projection functions of convex bodies. Arch. Math. 64 (1995), 69–74.
Bonnesen, T. and Fenchel, W., Theorie der konvexen Körper. Springer, Berlin 1934.
Gardner, R.J., Geometric Tomography. (Encyclopedia of Mathematics and its Applications, vol. 58), Cambridge University Press, Cambridge 1995.
Goodey, P., Schneider, R., Weil, W., On the determination of convex bodies by projection functions. Bull. London Math. Soc. 29 (1997), 82–88.
Goodey, P., Schneider, R., Weil, W., Projection functions of convex bodies. In Bolyai Society Math. Studies 6 (Intuitive Geometry, Budapest 1995, eds. I. Bárány and K. Böröczky sen), Budapest 1997, pp. 23 - 53.
Gruber, P.M., Baire categories in convexity. In Handbook of Convex Geometry (eds.P.M. Gruber and J.M. Wills), North-Holland, Amsterdam 1993, pp. 1327–1346.
Schneider, R., On the projections of a convex polytope. Pacific J. Math. 32 (1970), 799–803.
Schneider, R., Convex Bodies: the Brunn-Minkowski Theory. (Encyclopedia of Mathematics and its Applications, vol. 44), Cambridge University Press, Cambridge 1993.
Schneider, R., Polytopes and Brunn-Minkowski theory. In Polytopes — Abstract, Convex and Computational (eds. T. Bisztriczky, P. McMullen, R. Schneider and A. Ivić Weiss), NATO ASI Series, vol 40, Kluwer, Dordrecht, 1994, pp. 273–299.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schneider, R. On the determination of convex bodies by projection and girth functions. Results. Math. 33, 155–160 (1998). https://doi.org/10.1007/BF03322079
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322079