Let \(K\subset\mathbb{E}^{n+1}\) (\(n\geq 2)\) be a convex body, let \(p_ 0\in K\), and suppose that all \(n\)-sections through \(p_ 0\) are affinely equivalent. For odd \(n\), it is still unknown whether \(K\) must be an ellipsoid. The author proves the following weaker versions. If all \(n\)- sections of \(K\) through \(p_ 0\) are affinely equivalent, then either \(K\) is an ellipsoid or \(K\) is centrally symmetric with respect to \(p_ 0\). If all \(n\)-sections of \(K\) through \(p_ 0\) are volume-preserving affinely equivalent, then \(K\) is a Euclidean ball. (This extends a previous result of the reviewer [Bull. Lond. Math. Soc. 12, 52-54 (1980; Zbl 0401.52001)] on congruent sections.) If all \(n\)-sections of \(k\) through \(p_ 0\) are homothetic then \(K\) is a Euclidean ball.