We shall prove the following shaken Rogers's theorem for homothetic sections: Let $K$ and $L$ be strictly convex bodies and suppose that for every plane $H$ through the origin we can choose continuously sections of $K $ and $L$, parallel to $H$, which are directly homothetic. Then $K$ and $L$ are directly homothetic.

Keywords:

convex bodies, homothetic bodies, sections and projections, Rogers's Theorem convex bodies, homothetic bodies, sections and projections, Rogers's Theorem

MSC Classifications:

52A15 show english descriptions Convex sets in $3$ dimensions (including convex surfaces) [See also 53A05, 53C45] 52A15 - Convex sets in $3$ dimensions (including convex surfaces) [See also 53A05, 53C45]

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