Abstract
A convex set is inscribed into a rectangle with sides a and 1/a so that the convex set has points on all four sides of the rectangle. By “rounding” we mean the composition of two orthogonal linear transformations parallel to the edges of the rectangle, which makes a unit square out of the rectangle. The transformation is also applied to the convex set, which now has the same area, and is inscribed into a square. One would expect this transformation to decrease the perimeter of the convex set as well. Interestingly, this is not always the case. For each a we determine the largest and smallest possible increase of the perimeter.
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Communicated by Imre Bárány
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László, C. The perimeter of rounded convex planar sets. Period Math Hung 54, 31–49 (2007). https://doi.org/10.1007/s-10998-007-1031-2
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DOI: https://doi.org/10.1007/s-10998-007-1031-2