Characterizations of central symmetry for convex bodies in Minkowski spaces


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K. Zindler (47) and P. C. Hammer and T. J. Smith (19) showed the following: Let K be a convex body in the Euclidean plane such that any two boundary points p and q of K, that divide the circumference of K into two arcs of equal length, are antipodal. Then K is centrally symmetric. (19) announced the analogous result for any Minkowski planeM2, with arc length measured in the respective Minkowski metric. This was recently proved by Y. D. Chai{Y. I. Kim (7) and G. Averkov (4). On the other hand, for Euclidean d-space Rd, R. Schneider (38) proved that if K ‰Rd is a convex body, such that each shadow boundary of K with respect to parallel illumination halves the Euclidean surface area of K (for the deflnition of \halving" see in the paper), then K is centrally symmetric. (This implies the result from (19) for R2.) We give a common generalization of the results of Schneider (38) and Averkov (4). Namely, let Md be a d-dimensional Minkowski space, and K ‰Md be a convex body. If some Minkowskian surface area (e.g., Busemann's or Holmes{Thompson's) of K is halved by each shadow boundary of K with respect to paral- lel illumination, then K is centrally symmetric. Actually, we use little from the deflnition of Minkowskian surface area(s). We may measure \surface area" via any even Borel func-

Journal: Studia Scientiarum Mathematicarum Hungarica - STUD SCI MATH HUNG , vol. 46, no. 4, pp. 493-514, 2009

DOI: 10.1556/SScMath.2009.1104

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