Lassak (2002) showed that every convex set has axiality at least 2/3.
[1] This result improved a previous lower bound of 5/8 by Krakowski (1963).
[2] The best upper bound known is given by a particular convex quadrilateral, found through a computer search, whose axiality is less than 0.816.
[3] For triangles and for centrally symmetric convex bodies, the axiality is always somewhat higher: every triangle, and every centrally symmetric convex body, has axiality at least
-coordinates approach zero, showing that the lower bound is as large as possible.
It is also possible to construct a sequence of centrally symmetric parallelograms whose axiality has the same limit, again showing that the lower bound is tight.
[6] Barequet & Rogol (2007) consider the problem of computing the axiality exactly, for both convex and non-convex polygons.
The set of all possible reflection symmetry lines in the plane is (by projective duality) a two-dimensional space, which they partition into cells within which the pattern of crossings of the polygon with its reflection is fixed, causing the axiality to vary smoothly within each cell.
They thus reduce the problem to a numerical computation within each cell, which they do not solve explicitly.
cells for convex polygons; it can be constructed in an amount of time which is larger than these bounds by a logarithmic factor.
Barequet and Rogol claim that in practice the area maximization problem within a single cell can be solved in
[7] de Valcourt (1966) lists 11 different measures of axial symmetry, of which the one described here is number three.
Other symmetry measures with these properties include the ratio of the area of the shape to its smallest enclosing symmetric superset, and the analogous ratios of perimeters.
Lassak (2002), as well as studying axiality, studies a restricted version of axiality in which the goal is to find a halfspace whose intersection with a convex shape has large area lies entirely within the reflection of the shape across the halfspace boundary.
He shows that such an intersection can always be found to have area at least 1/8 that of the whole shape.
[1] In the study of computer vision, Marola (1989) proposed to measure the symmetry of a digital image (viewed as a function
from points in the plane to grayscale intensity values in the interval