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It is easy to prove that

(*) Every convex planar set of area 1 is contained in a quadrilateral of area 2.

It is also easy to see that statement (*) remains true if the constant 2 is replaced with a somewhat smaller one. Contest: Find such a constant, the smaller the better.

Update:

Reaching $\sqrt 2$ and even a strictly smaller value was proved by Chakerian (1973) and Kuperberg (1983) and the research challenge offered is to improve it even further, and perhaps even to verify the conjecture that the minimum is attained by a regular pentagon. But any nice arguments for bounds below 2 are welcome.

Community
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Wlodek Kuperberg
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1 Answers1

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G. D. Chakerian, Minimum area of circumscribed polygons, Elem. Math. 28 (1973), 108–111, MR0322682 (48 #1044) proved that if $K$ is a convex body of area 1 in the plane then $K$ is contained in a quadrilateral of area at most $\sqrt2$.

W. Kuperberg, On minimum area quadrilaterals and triangles circumscribed about convex plane regions, Elem. Math. 38 (1983), no. 3, 57–61, MR0703939 (85a:52009), proved that the infimum of the area ratio is strictly less than $\sqrt2$, and suggested the infimum might be attained when $K$ is a regular pentagon.

Gerry Myerson
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    The same W. Kuperberg who asked the question? – Robert Israel Jul 11 '13 at 04:14
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    I would hope not. I would hope that anyone posting a question here who had previously published on the question would remember what was published and include it in the statement of the question. In any event, even if my information is not news to OP, it may be news to others reading MO. – Gerry Myerson Jul 11 '13 at 05:55
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    @Gerry: Yes, it was me, and I remember. But I thought it would be fun for some people to try and see what they can discover on their own. Any explicit value below square root of 2 would be wonderful. And searching the internet for answers is fun, too, though probably not as much. – Wlodek Kuperberg Jul 11 '13 at 12:34
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    Wlodzimierz, please don't do anything like that again. – Scott Morrison Jul 11 '13 at 14:15
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    @Scott: Why not? Is it against the rules? Have I offended you or anyone else? Besides, even with Gerry's comments, the question remains valid. Any relevant remarks will be appreciated. – Wlodek Kuperberg Jul 11 '13 at 16:13
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    Prof. Kuperberg, the issue is not whether your question is good or bad; the issue is whether this is the place for it. MathOverflow has a purpose different from that of pursuing open problems: it is to get existing answers quickly to definite questions in a certain realm. Instead of presenting a chllenge, you might ask if there is recent progress, or if someone has tried a specific approach on a small part of the problem. – The Masked Avenger Jul 11 '13 at 16:39
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    Withholding from the problem question such relevant information as you possess (on what has been achieved) could cause would-be solvers a huge waste of time, and is if you ask me in rather bad taste. The information as provided in the update is vital and should never have been withheld in the first place. – Todd Trimble Aug 27 '13 at 01:25
  • Todd, I respectfully disagree... – Gil Kalai Sep 01 '13 at 16:38
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    Gil, what is it that you disagree with? – Todd Trimble Sep 02 '13 at 04:31
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    Todd, I disagree to the content, taste, tone, and timing of your Aug 27 1:25 comment. (If you wish to discuss it more please email me...) – Gil Kalai Sep 20 '13 at 16:43