Wrong Conjectures
Recording two conjectures and counter-examples.... Thanks to Prof. Erich Friedman.
1. Given any convex region C. Let R be the largest area rectangle that can be drawn inside C and R' the smallest area rectangle that can be drawn containing C. Claim: R and R' necessarily have the same alignment (same axes of symmetry). Note: A wider version of this claim could be stated generalizing from rectangles to any other convex shape with reflection symmetry.
Counter-example: Take a square, and shave off opposite corners at a 45 degree angle. For small cuts, the exterior square and interior square are optimal. For large cuts, a 45 degree rectangle is better. It would be a big surprise if the two phase transitions took place at exactly the same point.
And indeed, as can be checked numerically, the two phase transitions DO NOT happen at the same point.
o 2. Claim: The rectangle with largest perimeter that can be drawn inside any convex C is either degenerate (zero area, only length) or the same as the interior rectangle with highest area. Similarly, from among the rectangles containing C, the one with the least perimeter is the same as the one with least area. Note: This claim could be generalized from rectangles to ellipses or any other regions with reflection symmetry.
Counter: This claim is easily shown to fail by considering rectangles inscribed in any given ellipse. The inscribed rectangle with max perimeter is not usually the one with most area.
As for max containing rectangles, by considering the counterexample for claim 1( the progressively shaved square), we see that that for a certain stage in the shaving, the the 45 degree outer rectangle has less area but the outer square has less perimeter.
Note: Also tried to find CONTAINING ellipses with least area and least perimeter for a family of isosceles triangles all with same fixed apex and altitude but different bases. The least area containing ellipses of these triangles all have the same center (the centroid, which they all share anyways) - indeed these are their Steiner ellipses - but the centers of the least perimeter containing ellipses vary.
1. Given any convex region C. Let R be the largest area rectangle that can be drawn inside C and R' the smallest area rectangle that can be drawn containing C. Claim: R and R' necessarily have the same alignment (same axes of symmetry). Note: A wider version of this claim could be stated generalizing from rectangles to any other convex shape with reflection symmetry.
Counter-example: Take a square, and shave off opposite corners at a 45 degree angle. For small cuts, the exterior square and interior square are optimal. For large cuts, a 45 degree rectangle is better. It would be a big surprise if the two phase transitions took place at exactly the same point.
And indeed, as can be checked numerically, the two phase transitions DO NOT happen at the same point.
o 2. Claim: The rectangle with largest perimeter that can be drawn inside any convex C is either degenerate (zero area, only length) or the same as the interior rectangle with highest area. Similarly, from among the rectangles containing C, the one with the least perimeter is the same as the one with least area. Note: This claim could be generalized from rectangles to ellipses or any other regions with reflection symmetry.
Counter: This claim is easily shown to fail by considering rectangles inscribed in any given ellipse. The inscribed rectangle with max perimeter is not usually the one with most area.
As for max containing rectangles, by considering the counterexample for claim 1( the progressively shaved square), we see that that for a certain stage in the shaving, the the 45 degree outer rectangle has less area but the outer square has less perimeter.
Note: Also tried to find CONTAINING ellipses with least area and least perimeter for a family of isosceles triangles all with same fixed apex and altitude but different bases. The least area containing ellipses of these triangles all have the same center (the centroid, which they all share anyways) - indeed these are their Steiner ellipses - but the centers of the least perimeter containing ellipses vary.
posted by R.Nandakumar @ 3:29 AM
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