Convex bodies with pairs of sections associated by reflections

In this work we prove that if for a pair of convex bodies $K_1, K_2 \subset \mathbb{R}^n$, $n \geq 3$, there exists a hyperplane $H$ and two distinct points $p_1$ and $p_2$ in $\mathbb{R}^n \setminus H$ such that for every $(n-2)$-plane $M \subset H$, there exists a reflection mapping the hypersection of $K_1$ defined by $\mathrm{aff}{p_1, M}$ onto the hypersection of $K_2$ defined by $\mathrm{aff}{p_2, M}$, then there exists a reflection which maps $K_1$ onto $K_2$.

💡 Research Summary

The paper investigates a novel symmetry condition for pairs of convex bodies in Euclidean space of dimension $n\ge3$. Let $K_{1},K_{2}\subset\mathbb R^{n}$ be strictly convex, let $H$ be a hyperplane, and let $p_{1},p_{2}\in\mathbb R^{n}\setminus H$ be distinct. For every $(n-2)$‑dimensional subspace $M\subset H$ consider the affine hyperplanes $\pi_{i}(M)=\operatorname{aff}{p_{i},M}$, $i=1,2$. The hypothesis is that for each $M$ there exists a reflection $S_{M}$ with respect to the hyperplane containing $M$ such that \