Abstract
For a pair of convex bodies $K$ and $K’$ in $E^d$, the $d$-dimensional intersections $K \cap (x + K’),$ $x \in E^d,$ are centrally symmetric if and only if $K$ and $K’$ are represented as direct sums $K = R \oplus P$ and $K’ = R’ \oplus P’$ such that: (i) $R$ is a compact convex set of some dimension $m$, $0 \le m \le d,$ and $R’ = z - R$ for a suitable vector $z \in E^d$, (ii) $P$ and $P’$ are isothetic parallelotopes, both of dimension $d-m$.
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Soltan, V. Pairs of Convex Bodies with Centrally Symmetric Intersections of Translates. Discrete Comput Geom 33, 605–616 (2005). https://doi.org/10.1007/s00454-004-1094-6
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DOI: https://doi.org/10.1007/s00454-004-1094-6