Sections and projections of homothetic convex bodies

Abstract.

For a pair of convex bodies K ₁ and K ₂ in Euclidean space $\mathbb{E}^n$, n ≥ 3, possibly unbounded, we show that K ₁ is a translate of K ₂ if either of the following conditions holds: (i) the orthogonal projections of K ₁ on 2-dimensional planes are translates of the respective orthogonal projections of K ₂, (ii) there are points p ₁ ∈K ₁ and p ₂ ∈K ₂ such that for every pair of parallel 2-dimensional planesL ₁and L ₂ through p ₁ and p ₂, respectively, the section K ₁ ∩ L ₁is a translate of K ₂ ∩ L ₂.