Reference request: Where does this proposition about ellipses appear in the literature?

Question

In this question I conjectured that a simple proposition about ellipses holds. In the accepted answer, "Chrystomath" proves it.

Is this in some published refereed source?

quote:

Consider a closed bounded set with non-empty interior in the plane. Suppose it is strictly convex, i.e. every point between two of its points is one of its interior points. This entails that a line that intersecting its boundary but not its interior intersects it at only one point. Call such a line a tangent line. It follows that for every tangent line, there is exactly one other tangent line parallel to it. Suppose that for every line parallel to those two and between them, the midpoint of the intersection of that line with our closed bounded convex set is on the line connecting the two points of tangency.

Does it follow that our closed bounded set is the convex hull of an ellipse?

Answer 1

Posted and answered on mathoverflow.net

Yes, this characterization is a theorem proven by Blaschke in "Kreis und Kugel" (1916). The theorem has a higher dimensional version, characterizing ellipsoids as the unique strongly convex bodies with the property that all centroids of codimension 1 sections that are parallel to a fixed plane lie on a line. See the paper "Characterizations of ellipsoids by section-centroid location" by Meyer and Reisner.