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One up-vote, 35 views, and no comments and no answers have resulted from this reference request that I posted on math.stackexchange.com . This was actually inspired by a probability exercise, and at this point I'm half-way expecting someone to figure out how that happened before anyone finds the answer. But we'll see.

(Note that there are two links to m.s.e. in this present question. The one that appears above is a link to the reference request; the one below is to an earlier posting.)


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.

[i.e. for EVERY such pair of tangent lines]

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

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  • $\begingroup$ The probability exercise was stated like this: Suppose $X,Y$ are real-valued random variables and for some numbers $a,b$ with $a\ne 0$ we have $\operatorname E(Y\mid X) = aX+b.$ Prove or disprove that from that alone it follows that for some numbers $c,d$ with $c\ne0$ we have $\operatorname E(X\mid Y) = cY+d.$ That was the exercise. It is well known that if $(X,Y)$ is jointly normally distributed, i.e. so distributed that every linear combination $pX+qY$ is normally distributed,$\,\ldots\qquad$ $\endgroup$ Aug 28, 2020 at 20:36
  • $\begingroup$ $\ldots\,$then both of the conditional expectations are as above (with $c=1/a$ ONLY IF $\operatorname{cor}(X,Y)\in\{\pm1\}$), and that the level sets of the joint density are ellipses. And it's easy to see that with any joint density whose level sets are ellipses we get those two straight lines. Thus counterexamples are to be found only among joint densities whose level sets are not ellipses. But is EVERY instance where they're not ellipses a counterexample? $\qquad$ $\endgroup$ Aug 28, 2020 at 20:38

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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.

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  • $\begingroup$ BEGIN QUOTE $${}$$ In [2] Blaschke proved the following result which he relates to Brunn: Let $K$ be a convex body in $\mathbb R^n$ with the following property: the midpoints of every bundle of parallel chords of $K$ lie in a plane, then (and only then) $K$ is an ellipsoid. $${}$$ END QUOTE $${}$$That's the very first sentence in that paper. (The "only then" part is trivial.) $\qquad$ $\endgroup$ Aug 29, 2020 at 19:30
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    $\begingroup$ Here's a small data point that supports my preference for the more traditional way of using hyphens, which calls for using them somewhat more often than is often done today: "centroids of codimension 1 sections" For a second I thought: What is a centroid of codimension 1? Then I realized that "codimension 1" is an adjective qualifying the word "sections". So "centroids of codimension-1 sections" is how I would write it. Richard Borcherds once expressed the fear that such a thing could be mistaken for a minus sign, but I don't think that's true with proper typesetting. $\endgroup$ Aug 29, 2020 at 19:48

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