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 A {\displaystyle A} and perimeter P {\displaystyle P} , containing N {\displaystyle N} integer points,
A − 1 2 P < N ≤ A + 1 2 P + 1. {\displaystyle A-{\frac {1}{2}}P<N\leq A+{\frac {1}{2}}P+1.}
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 N = 0 {\displaystyle N=0} was later rediscovered by Edward Bender. The same special case has been generalized (with a larger constant than 1 2 {\displaystyle {\tfrac {1}{2}}} ) 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 A {\displaystyle A} by the volume and P {\displaystyle P} by the surface area of the set.
References
External links Weisstein, Eric W., "Nosarzewska's Inequality", MathWorld
