Scientiae Mathematicae Japonicae
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The Convexity Ratio and Applications
James R. Bozeman and Matthew Pilling
Abstract. We introduce the convexity ratio, CR(P), which indicates how close a
simple closed polygon P is to being convex. This is accomplished by considering the
ratio of the area of a largest convex set (the endogon ) contained in P to the area of the
convex hull of P. Algorithms for determining the convex hull and for determining the
endogon are exhibited. After defining when such polygons are nearly convex we then
use this result to decide when legislative districts are nicely shaped. We show that this
method for measuring the shape of legislative districts is better than, or as good as,
other techniques in the literature. This is accomplished by pointing out flaws in the
other methods and by examining examples of district shapes found in the literature.
1 Introduction The convex hull of a set of points in the plane is a well-defined mathematical object (see the next section for definitions). If given n points in the plane then there
exist algorithms for finding the convex hull, or exogon, of these points in nlogn time (see,
e.g., Graham [5]). Note that the convex hull in these cases will be polygonal. A more recent
problem is to find the largest convex polygon contained inside a given simple polygon, the
endogon. This is the potato - peeling problem of Goodman [4]. Such a polygon exists (see
below for details) but this polygon is not well-defined. Goodman [4] gave a finite solution
to this problem if the polygon has ≤ 5 sides. He also exhibited some properties of solutions
to the general problem. Chang and Yap [3] give an O(n7 ) time algorithm for finding this
set, solving this problem computationally.
The redrawing of legislative districts in each state in the United States is an ongoing
and controversial issue. Population numbers and the demographics of the population are
taken into account in redistricting, as is the geography of the state. The political party in
power at the time of the redistricting also tries to draw districts while taking into account
the political affiliation of members of the district, to ensure that more members of the
party in power are elected. This has lead to districts which meander around the state, are
elongated and are very jagged. In general, they are not nicely shaped . In an attempt to
take politics out of this redrawing, and to have districts that are as nicely shaped as possible,
mathematical formulae have been developed to assign numerical values to district shapes.
Generally, the closer to 1 this value is, the more nicely shaped it is. As this number gets
closer and closer to zero, the uglier the shape becomes. Some of these tests measure what
Schwartzberg [13] termed the ‘compactness’ of the district. To a mathematician, of course,
all of the districts are compact , but this terminology has stuck in the literature. All of
these tests really speak to the convexity of a district and this is what our new test utilizes.
Our results in this paper are organized as follows: In Section 2 we present the definitions
and terminology and give preliminary results. In Section 3 we apply this work to measuring
the shape of legislative districts. We show that our method is better than, or as good as, the
other methods in the literature. Section 4 discusses computational techniques and includes
a discussion of the next steps in this study. We conclude in Section 5.
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This work was supported in part by grants through Lyndon State College (LSC). The
authors were partially supported by the Brierley (STAR) Endowment Fund and the Burnham Faculty Development Fund. The second author researched this topic while an undergraduate at LSC. The first author has been supported by numerous Lyndon State College
Advanced Study Grants.
2 Definitions, Terminology and Preliminary Results We work throughout this
paper in the Euclidean plane R2 .
Definition 1: A simple polygonal region in the plane is a compact set in the plane
bounded by a simple closed polygon.
Definition 2: A region (set of points) in the plane is said to be convex if every pair
of points in the region can be connected by a straight line lying entirely in the region.
In what follows, see Bozeman, et al [1] for the details:
Definition 3: The convex hull or exogon of a simple polygonal region in the plane
is the smallest convex set in the plane containing the polygon. (That this is a well-defined
polygon is a well-known result.)
Definition 4: An endogon is a largest convex set contained in a simple polygonal
region in the plane.
That an endogon exists for every simple polygonal region in the plane follows from:
Proposition 1: If C is a compact set in the plane, then C contains a closed convex
subset of maximum area.
That an endogon is itself a simple polygonal region follows from:
Proposition 2: If M is any maximal convex subset of a simple polygonal region in the
plane, then M itself is a simple polygonal region.
See [1] for an example showing an endogon is not well-defined, in particular not unique.
Definition 5: The convexity ratio of a simple polygonal region P in the plane is
denoted CR(P) = (area of the endogon of P)/(area of the exogon of P).
Definition 6: A simple polygonal region P in the plane is said to be nearly convex if
CR(P) ≥ 0.5.
In Figure 1 we have set CR(P) = 0.5.
3 Nicely Shaped Legislative Districts Numerous mathematical measures have been
introduced which assign numbers to the shape of legislative districts in order to decide if the
district is well-shaped or not. These tests give values as to the ‘compactness’ of a district.
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Figure 1:
This is not mathematical compactness, however, but the English definition of ‘compact’.
Some of these tests are:
Schwartzberg Test [13]: The ratio of the legislative district’s perimeter to its area. (Note
that this value is not necessarily between 0 and 1.)
Polsby-Popper Test [10]: The ratio of the area of the legislative district to the area A
of a circle with circumference equal to the perimeter P of the district. This is given by the
formula (4πA)/(P 2 ) (verification left to the reader). This test corrects a problem with the
Schwartzberg test (see below) and produces numbers between 0 and 1.
Roeck Test [12]: The ratio of the area of the legislative district to the area of the smallest
circle containing the district. This value is also between 0 and 1.
Each of these measures have difficulties. The Schwartzberg test gives varying values to
districts, so it is hard to choose a numerical cut-off. Even worse, perfectly shaped districts
yield different values. For instance, a square with sides 2 miles has a value of 2, while a
square with sides 10 miles has a value of 0.4. This is certainly undesirable. The Roeck test
has a different problem. It requires the area of the circumcircle of the district. While there
is a linear time algorithm for finding this (see Megiddo [8]), the circumcircle can have an
extremely large area compared to the exogon, meaning the ratio obtained will be artificially
small. (See Figure2 below.)
One of the most popular and most used methods is the Polsby-Popper test. It outputs
values between 0 and 1 and is the ratio of the area of the district, which is known, to the area
of the circle with circumference equal to the perimeter of the district, the latter value also
being known. So it is easy to calculate and allows one to choose a cut-off value. However,
this test does not always yield values corresponding to how nicely a district is shaped, since
the perimeter can be arbitrarily long. This can be seen in the following figure and what
follows:
The district on the right is clearly better shaped than the one on the left. Nonetheless,
the Polsby-Popper test gives both these districts values < 0.5. In fact the district on the
right has Polsby- Popper value 0.411 while the one on the left has value 0.337, quite close
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Figure 2:
to one another. The Roeck test provides even worse values: the district on the right has a
value of 0.33, while the one on the left is 0.38. So the region with the nicer shape has a lower
Roeck value! Now the values of CR(P) for these districts are 0.449 and 0.848, respectively.
(Note that these latter values are approximations, albeit accurate ones. One of the nice
things about our technique is that the endogon can be ‘eyeballed’ in most cases.) We have:
Definition 7: A legislative district is nicely shaped if it is nearly convex. Otherwise,
it is poorly shaped.
Then the figure on the right is nicely shaped, as expected, while the figure on the left is
poorly shaped, again as expected.
We now examine numerous examples in the literature, which show that the convexity
ratio is better than, or at least as good as, current ‘compactness’ measures. The next figure
is of a polygon which is nicely shaped and should be calculated as such:
Figure 3:
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This polygon, while not convex, is a reasonable building block for a nicely shaped district.
(The same can be said of Figure 1) The convexity ratio for this figure is 0.57, as expected.
However, the Roeck test yields 0.48.
The following two polygons are poorly shaped and again should have measures indicating
this:
Figure 4:
In this case the convexity ratio yields 0.45, as expected. But Roeck’s test, counterintuitively, produces 0.59.
Figure 5:
In this clearly poorly shaped polygon, CR(P) = 0.25, Roeck = 0.56!
4 Computational Matters and Next Steps There are many well-known algorithms
for finding the exogon, i.e. the convex hull, of a simple closed polygonal region. The best
of these take O(nlogn) time, where n is the number of coordinates in the plane. One is by
Graham [5], who was the first to achieve nlogn time. There are also many programs which
will calculate the area of the exogon. One such is written in MATLAB and can be accessed
at http://www.mathworks.com/help/matlab/ref/ delaunaytri.convexhull.html.
An O(n7 ) time algorithm for finding the endogon can be found in Chang and Yap [3].
In this paper they show that the problem is finite, in general, and derive a polynomial time
algorithm. They introduce the concept of a balanced chain to accomplish this and it turns
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out this concept is the stumbling block in finding a faster approach (we are not aware of a
quicker algorithm). The algorithm itself is difficult (if not impossible) to program, given its
theoretical nature, and it does not calculate the area of the endogon. Goodman, in [4], gave
finite solutions in the n = 4 and n = 5 cases. He also provided some properties of solutions
to the general problem.
The first author and undergraduate students at LSC are currently working on a program,
in MATLAB, which will calculate the endogon and its area in the n = 4 and n = 5 cases. We
are also working on a program to best approximate the area of the endogon in the general
case. The goal is to be able to quickly decide when a proposed (or current) legislative
district is nicely shaped or not, i.e. to determine whether or not a simple polygonal planar
set is nearly convex.
5 Conclusion We agree with Horn, et al [7], that ‘...a measure must be understandable
and computable by the general public.’ (We take this to include legislators.) We disagree
with Polsby-Popper [11], who state that their measure, which is related to Schwartzberg’s
[13], should be used ‘Despite any theoretical objection to the Schwartzberg criterion’. Any
measure utilized should be valid in most, if not all, cases and it is clear from the examples
above that the Schwartzberg criterion does not meet this threshold. So we agree with Horn,
et al [7], Niemi, et al [9] and, especially, Young [14] who states that ‘ “compactness” is not
a sufficiently precise concept to be used as a legal standard for redistricting plans.’ Hence
the convexity ratio should be used when measuring the shape of legislative districts.
Chambers and Miller in [2] use a path-based measure of convexity which calculates the
probability that a district will contain the shortest path between a randomly selected pair
of points. See also Hodge, et al [6] for a ‘simpler’ version of this measure, which they call
the convexity coefficient. But as the authors in [6] indicate, ‘Calculating the exact value
of the convexity coefficient of a region can be difficult, even for relatively simple shapes’.
Therefore this would be hard for the public, and legislators, to understand.
What follows is what appears to be a gerrymandered district, the 4th district of Illinois,
together with its convexity ratio and its convexity coefficient, indicating our measure is as
good as (if not better than) those of Chambers and Miller and of Hodge, et al.
Figure 6:
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CR(P) = 0.234, convexity coefficient = 0.237
Many considerations must be taken into account when redrawing legislative districts.
Some of these include the number of voters in the proposed district, the voters’ ethnicity
and their (supposed) shared interests. If the shape of the legislative district is to be a factor,
and if a mathematical measure is to be employed to decide if a district is nicely shaped or
not, then the convexity ratio of the district should be employed.
References
[1] J.R. Bozeman, L. Pyrik, J. Theoret, Nearly Convex Sets and the Shape of Legislative Districts,
Int. Jour. Pure and Appl. Math. 49 (2008), 341-347.
[2] C.P. Chambers, A.P. Miller, A Measure of Bizarreness, (2008) (preprint).
[3] J.S. Chang, C.K. Yap, A Polynomial Solution for the Potato-peeling Problem, Discrete Comput.
Geom. 1 (1986), 155-182.
[4] J.E. Goodman, On The Largest Convex Polygon Contained In A Non-convex n-gon, Or How
To Peel A Potato, Geometriae Dedicata 11 (1981), 99-106.
[5] R.L. Graham, An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set,
Information Processing Letters 1 (1972), 132-133.
[6] J.K. Hodge, E. Marshall, G. Patterson, Gerrymandering and Convexity, College Mathematics
Journal 41(4) (2010), 312-324.
[7] D.L. Horn, C.R. Hampton, A.J. Vandenburg, Practical Applications of District Compactness,
Political Geography 12 (1993), 103-120.
[8] N. Megiddo, Linear-time algorithms for linear programming in R3 and related problems, SIAM
Journal on Computing 12 (1983), 759-776.
[9] R.G. Niemi, B. Grofman, C. Carlucci, T. Hofeller, Measuring compactness and the role of
a compactness standard in a test for partisan and racial gerrymandering, Journal of Politics
52(4) (1990), 1155-1181.
[10] D.D. Polsby, R.D. Popper, The Third Criterion: Compactness as a Procedural Safeguard
Against Partisan Gerrymandering, Yale Law and Policy Review 9 (1991), 301-353.
[11] D.D. Polsby, R.D. Popper, Partisan Gerrymandering: Harms and a New Solution, Heartland
Policy Studies 34 (1991).
[12] E.C. Roeck, Jr., Measuring the Compactness as a Requirement of Legislative Apportionment,
Midwest Journal of Political Science 5 (1961), 70-74.
[13] J.E. Schwartzberg, Reapportionment, Gerrymanders, and the Notion of Compactness, Minnesota Law Review 50 (1966), 443-452.
[14] H.P. Young, Measuring the compactness of legislative districts, Legislative Studies Quarterly
13(1) (1988), 105-115.
James R. Bozeman: Mathematics and Computer Science, Lyndon State College,
1001 College Road, Lyndonville, VT 05851, USA,
Email: [email protected]
Matthew Pilling: Dept. of Mathematics, Univ. of Rhode Island, Kingston, RI
02881, USA,
Email: matthew [email protected]