Biggest Little Polygon
The biggest little polygon with 
sides is the convex plane 
-gon of unit polygon diameter having largest possible area.
Reinhardt (1922) showed that for 
odd, the regular polygon on 
sides is the biggest little 
-gon. For 
, the square with diagonal 1 has maximum area, but an infinite number of other 4-gons are equally large (Audet et al. 2002). The 
case was solved by Graham (1975) and is known as Graham's biggest little hexagon, and the 
case was solved by Audet et al. (2002). The following table summarizes these results, showing the percentage that the given polygon is larger than the regular 
-gon.
 |
area |
% larger than regular  -gon |
reference |
| 6 |
0.674981 |
3.92% |
Graham (1975) |
| 8 |
0.726867 |
2.79% |
Audet et al. (2002) |
The biggest little polygon graphs on 
and 8 nodes are implemented in the Wolfram Language as GraphData[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/BiggestLittlePolygon/Inline13.gif" style="height:15px; width:5px" />"BiggestLittlePolygon", n![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/BiggestLittlePolygon/Inline14.gif" style="height:15px; width:5px" />].
REFERENCES:
Audet, C. "Optimisation globale structurée: propriétés, équivalences et résolution." Thèse de Doctorat. Montréal, Canada: École Polytechnique de Montréal, 1997. http://www.gerad.ca/Charles.Audet.
Audet, C.; Hansen, P.; Messine, F.; and Xiong, J. "The Largest Small Octagon." J. Combin. Th. Ser. A 98, 46-59, 2002.
Graham, R. L. "The Largest Small Hexagon." J. Combin. Th. Ser. A 18, 165-170, 1975.
Reinhardt, K. "Extremale Polygone gegebenen Durchmessers." Jahresber. Deutsch. Math. Verein 31, 251-270, 1922.
