On the girth of convex bodies (Q1364587)

Let \(K_{E}\) be the orthogonal projection of a convex body \(K\subset\mathbb{R}^{3}\) onto a plane \(E\) through the origin. The perimeter of \(K_{E}\) is called the girth of \(K\) with respect to \(E.\) It is known that the girth of \(K\), as a function of \(E\), does not determine the body \(K\) up to translations, but convex bodies having the same girth are equiwide [see \textit{S. Nakajima}, Jap. J. Math. 7, 225-226 (1930; JFM 56.1204.02)]. For any unit vector \(\mathbf{u}\) in \(E\) that is a regular direction of \(K_{E}\), the ``upper'' arc with respect to \(\mathbf{u}\) is one of the two arcs of \(K_{E}\) with the support points in the direction \(\mathbf{u}\) as endpoints. The length of this ``upper'' arc is called the semi-girth of \(K\) corresponding to \(E\) and \(\mathbf{u}\). The author proves that two convex bodies having the same semi-girth, as a function of \(E\) and \(\mathbf{u}\), are translates of each other. This result is a consequence of a more general stability result on semi-girths. Moreover he establishes an inequality concerning the central symmetry of convex domains.