In the geometry of numbers, Nosarzewska's inequality relates the area and perimeter of a two-dimensional convex set with the number of integer lattice points that it contains. The inequality states that, for a convex set of area and perimeter , containing integer points,
This implies that when measuring the area of a convex set using a dot planimeter, the error in estimation is at most proportional to the perimeter. The theorem is named after Maria Nosarzewska, a student of Polish mathematician Edward Marczewski; Nosarzewska published it in 1948. In her paper on the subject, Nosarzewska writes that she was answering a question posed by Hugo Steinhaus.
A special case of the inequality for the case was later rediscovered by Edward Bender. The same special case has been generalized (with a larger constant than ) to regions bounded by Jordan curves. Nosarzewska's inequality has also been generalized to convex sets in higher dimensions, with inequalities of the same form that replace by the volume and by the surface area of the set.
References ; see p. 165
External links