An asymmetric convex body with maximal sections of constant volume (Q2862634)

Let \( K \subset {\mathbb R}^d\) be a convex body containing the origin in its interior. Its \textit{maximal section function} is defined as NEWLINE\[NEWLINE M_K (u) = \max_{t \in {\mathbb R}} \,\text{vol}_{d-1} (K \cap (u^{\bot} + t u)), \quad u \in {\mathbb S}^{d-1} . NEWLINE\]NEWLINE Here \( u^{\bot} + t u\) is the affine hyperplane orthogonal to the unit vector \( u \).NEWLINENEWLINE The main result of the paper isNEWLINENEWLINETheorem 1. If \( d \geq 3 \), there exists a convex body of revolution \( K \subset {\mathbb R}^d\) satisfying \( M_K \equiv \text{const}\) that is not an Euclidean ball.NEWLINENEWLINEThis is a negative answer to a question of \textit{V. Klee} [``Is a body spherical if its HA-measurements are constant?'', Am. Math. Monthly 79, No. 5, 539--542 (1969; \url{doi:10.2307/2316970})].NEWLINENEWLINE The proof of the theorem is quite different in the cases of odd \( q \) and even~\( q \), and is based on analysis of a system of integral equations combined with some fine hand-made geometrical constructions.