Abstract
We investigate the problem of finding the maximum length of perimeters of plane sets with fixed diameter d, such that every point of the boundary of the set is a vertex of an open angle of opening α which does not intersect the set. First we consider plane curves which satisfy such angle property in a finite number of directions, and among them we find the one of maximum length. Then we prove that the perimeter of any plane set with the angle property is less than or equal to πd(sin α/2)-2; this is the best estimate when π/2≤α≤π.
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References
Gerver, J. L.: On moving a sofa around a corner, Geom. Dedicata 42 (1992), 267–283.
Saroldi, M.: Sulla regolarità di frontiere di insiemi con proprietà di cono, Degree Thesis, Univ. of Florence, 1993.
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Saroldi, M. On the best estimate for perimeters of plane sets with the angle property. Geom Dedicata 63, 193–204 (1996). https://doi.org/10.1007/BF00148219
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DOI: https://doi.org/10.1007/BF00148219