Nosarzewska's inequality

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 $A$ and perimeter $P$ , containing $N$ integer points,^[1] $A-{\frac {1}{2}}P<N\leq A+{\frac {1}{2}}P+1.$ 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;^[2] Nosarzewska published it in 1948. In her paper on the subject, Nosarzewska writes that she was answering a question posed by Hugo Steinhaus.^[3]

A special case of the inequality for the case $N=0$ was later rediscovered by Edward Bender.^[4] The same special case has been generalized (with a larger constant than ${\tfrac {1}{2}}$ ) to regions bounded by Jordan curves.^[5] Nosarzewska's inequality has also been generalized to convex sets in higher dimensions, with inequalities of the same form that replace $A$ by the volume and $P$ by the surface area of the set.^[6]^[7]

References

↑ Apostol, Tom M. (2000), "Lattice points", Cubo Matemática Educacional, 2: 157–173, MR 1957370; see p. 165
↑ Duda, Roman; Hartman, Stanisław, "Edward Marczewski obituary", MacTutor History of Mathematics Archive, translated by Iwanik, A.; Lipecki, Z., retrieved 2026-07-08
↑ Nosarzewska, M. (1948), "Évaluation de la différence entre l'aire d'une région plane convexe et le nombre des points aux coordonnées entières couverts par elle", Colloquium Mathematicum (in French), 1 (4): 305–311, doi:10.4064/cm-1-4-305-311, MR 0030551
↑ Bender, Edward A. (1962), "Area-perimeter relations for two-dimensional lattices", American Mathematical Monthly, 69 (8): 742–744, doi:10.1080/00029890.1962.11989969, JSTOR 2310769, MR 1531832
↑ Schnell, Uwe; Segura Gomis, Salvador (1996), "Two problems concerning the area-perimeter ratio of lattice-point-free regions in the plane", Beiträge zur Algebra und Geometrie, 37 (1): 1–8, MR 1407800
↑ Bokowski, Jürgen; Hadwiger, Hugo; Wills, Jörg M. (1972), "Eine Ungleichung zwischen Volumen, Oberfläche und Gitterpunktanzahl konvexer Körper im $n$ -dimensionalen euklidischen Raum", Mathematische Zeitschrift (in German), 127 (4): 363–364, doi:10.1007/BF01111393, MR 0315595
↑ Bokowski, J.; Odlyzko, A. M. (1973), "Lattice points and the volume/area ratio of convex bodies", Geometriae Dedicata, 2 (2): 249–254, doi:10.1007/BF00147862, MR 0420438

External links

Weisstein, Eric W., "Nosarzewska's Inequality", MathWorld

[apostol-1] Apostol, Tom M. (2000), "Lattice points", Cubo Matemática Educacional, 2: 157–173, MR 1957370; see p. 165

[edmar-2] Duda, Roman; Hartman, Stanisław, "Edward Marczewski obituary", MacTutor History of Mathematics Archive, translated by Iwanik, A.; Lipecki, Z., retrieved 2026-07-08

[nosarzewska-3] Nosarzewska, M. (1948), "Évaluation de la différence entre l'aire d'une région plane convexe et le nombre des points aux coordonnées entières couverts par elle", Colloquium Mathematicum (in French), 1 (4): 305–311, doi:10.4064/cm-1-4-305-311, MR 0030551

[bender-4] Bender, Edward A. (1962), "Area-perimeter relations for two-dimensional lattices", American Mathematical Monthly, 69 (8): 742–744, doi:10.1080/00029890.1962.11989969, JSTOR 2310769, MR 1531832

[ssg-5] Schnell, Uwe; Segura Gomis, Salvador (1996), "Two problems concerning the area-perimeter ratio of lattice-point-free regions in the plane", Beiträge zur Algebra und Geometrie, 37 (1): 1–8, MR 1407800

[bhw-6] Bokowski, Jürgen; Hadwiger, Hugo; Wills, Jörg M. (1972), "Eine Ungleichung zwischen Volumen, Oberfläche und Gitterpunktanzahl konvexer Körper im $n$ -dimensionalen euklidischen Raum", Mathematische Zeitschrift (in German), 127 (4): 363–364, doi:10.1007/BF01111393, MR 0315595

[bokod-7] Bokowski, J.; Odlyzko, A. M. (1973), "Lattice points and the volume/area ratio of convex bodies", Geometriae Dedicata, 2 (2): 249–254, doi:10.1007/BF00147862, MR 0420438

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