# About the 'minimum triangle' which includes a convex bounded closed set

Question : Is the following true?

"Letting $K$ be a convex bounded closed set on a plane, then there exists a triangle $M$, which includes $K$, such that $|M|\le 2|K|$. Here, $|M|,|K|$ is the area of $M,K$ respectively."

Motivation : First, I've thought about the case that $K$ is a parallelogram. Then, I reached the above expectation, but I can neither prove this nor find any counterexample. Can anyone help?

Remark : If $K$ is a parallelogram, then $|M|\ge2|K|$ for any $M$ which includes $K$. This question has been asked previously on math.SE without receiving any answers.

## 1 Answer

MathWorld attributes this theorem to Gross, 95 years ago:

Gross, W. "Über affine Geometrie XIII: Eine Minimumeigenschaft der Ellipse und des Ellipsoids." Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Math.-Phys. Kl. 70, 38-54, 1918.

(Added). Here is a figure from Wlodek's article, cited in the comments: • Moreover, one of the sides of the desired triangle $M$ can be parallel to an arbitrarily chosen line, see: Kuperberg, W., On minimum area quadrilaterals and triangles circumscribed about convex plane regions. Elem. Math. 38 (1983), no. 3, 57–61. MR0703939 (85a:52009) – Wlodek Kuperberg Oct 28 '13 at 21:44
• Here's a link to Wlodek's article: journal link. – Joseph O'Rourke Oct 28 '13 at 21:58