Convexity, gerrymandering
Is there a word for a Euclidean geometric figure that is such that one can draw a straight line from any point in it to any other point in it?
Yes. In geometry, the property you're describing is called convexity.
A set (or figure) is convex if, for every pair of points inside the figure, the entire line segment joining them also lies inside the figure.
That's exactly your rule:
If you can draw a straight line from any point in the figure to any other point in the figure, and that line never leaves the figure, then the figure is convex.
Examples of convex figures:
Triangle
Rectangle
Square
Circle
Ellipse
Regular hexagon
Examples of non-convex (concave) figures:
Crescent shape
Star shape
Many "L"-shaped regions
For example, in an L-shaped figure, you can choose two points inside it such that the straight line between them passes through empty space outside the figure. That violates convexity.
A useful theorem is that a polygon is convex if and only if all its interior angles are less than 180°.
This idea turns out to be important far beyond Euclidean geometry—in economics, optimization, computer graphics, and even anthropology-related questions about territory and movement. Convex shapes are, in a sense, the shapes with no "indentations" or "caves" in them.
I have in mind an application in politics, as a rule in drawing districts to make gerrymandering more difficult. Issues?
Your idea has a long pedigree. In fact, what you described—requiring districts to be convex—is one of the more mathematically elegant anti-gerrymandering proposals.
The attraction is obvious:
A convex district cannot have the classic "tentacles" and "salamander tails" associated with gerrymanders.
The rule is objective and easy to explain to voters.
Whether a district is convex is a yes/no question, not a matter of degree.
Computers can test it instantly.
However, there are several issues.
1. Convexity is often too strict
Real geography is messy.
A coastal district with a bay cut into it may be non-convex through no fault of the mapmaker. The same applies to districts constrained by rivers, mountains, state borders, or irregular municipal boundaries.
A district around a large lake is almost guaranteed to violate strict convexity.
2. Convex districts can still be gerrymandered
This is the surprising part.
Researchers have shown that even if every district must be convex, substantial partisan advantage can still be achieved under many voter distributions. If one party's voters are geographically clustered, clever partitioning with entirely convex districts can still "pack" or "crack" voters.
So convexity limits the most grotesque shapes, but does not eliminate the underlying problem.
3. Representation of communities
Suppose a minority community occupies a crescent-shaped urban corridor following a river.
A convexity requirement might force that community to be split among several districts, reducing its ability to elect a representative of choice.
This is one reason redistricting systems often include concepts such as "communities of interest" alongside geometric criteria.
4. Population equality complicates things
If districts must contain equal populations, perfect convexity may be impossible.
Imagine a state where most people live in a few cities. Drawing several equal-population convex districts may require boundaries that look mathematically neat but socially strange.
5. Convexity versus compactness
Most reform proposals use compactness rather than strict convexity.
Compactness asks whether districts are reasonably "tight" or "not sprawling." Convexity is one way to measure that, but not the only way. Political scientists have proposed dozens of compactness measures because no single geometric rule captures all notions of fairness.
An anthropological observation
Your proposal reminds me of something much older than modern elections.
Many traditional territorial divisions—village lands, grazing territories, parish boundaries—tend naturally toward convexity because travel and communication are easier when any two points can be connected without leaving the territory. Modern districting inherits that intuition.
The difficulty is that elections are not merely about territory. They are also about populations, communities, ethnic groups, cities, counties, and political preferences. Once those factors enter, geometry alone ceases to be enough.
If I were designing a system from scratch, I would be more inclined toward:
Equal population.
Contiguity (all parts connected).
High compactness (not necessarily strict convexity).
Respect for county/city boundaries where practical.
Districts drawn by an independent algorithm or commission rather than legislators.
That tends to attack the incentive for gerrymandering rather than relying on a single geometric rule. Convexity is attractive because it is clean mathematics, but politics has a way of finding loopholes in even very clean mathematics.
Number four is inappropriate to my mind, just an excuse to make exceptions.
There's a reasonable argument for that view.
One of the recurring problems in redistricting is that every additional criterion creates another opportunity for discretion. "Respect county boundaries," "preserve communities of interest," "maintain historical districts," and similar rules can all be sincere attempts at fairness, but they can also become justifications for drawing almost any map someone wants.
From that perspective, a small set of simple rules has a lot going for it:
Equal population.
Contiguity.
Compactness (or even convexity).
No consideration of voting history, party registration, incumbents, race, etc., except where required by law.
The appeal is that anyone can understand the rules and verify them.
The counterargument is that political boundaries sometimes correspond to real administrative units. If a county government, school system, or city government serves a population, splitting that population among many districts may reduce accountability or make representation more confusing. Whether that benefit outweighs the opportunity for manipulation is a judgment call.
There is also a deeper mathematical issue. Even if you impose convexity and ignore county lines entirely, there is usually no unique "correct" map. A state can often be partitioned into equal-population convex districts in many different ways. If humans are choosing among those possibilities, the opportunity for bias remains.
That is why some reformers eventually arrive at an even more radical position: don't ask people to draw districts at all. Define a mechanical procedure and let a computer generate the map from census data. Then the fight shifts from "Which map?" to "Which algorithm?"
In a sense, convexity is attractive because it is trying to turn a political problem into a geometric one. The difficulty is that geometry narrows the range of possible maps but rarely determines a unique answer. As you often point out in other contexts, once humans have discretion, they tend to find ways to use it.
Local administration is one thing, representation in Congress another. They are not linked. Having a computer do it with an independent body handling the matter is about the best one can do.
That's a position many electoral reformers would recognize, though they differ on the details.
Your distinction between administration and representation is important. A county, city, or parish exists because government functions have to be administered somewhere. A congressional district exists for a different purpose: allocating representation in a legislature. There is no inherent reason the two sets of boundaries must coincide.
Historically, many political systems inherited electoral districts from older administrative units because they already existed and people knew them. But that's more a matter of convenience and tradition than logical necessity.
The attraction of an algorithmic approach is that it removes much of the incentive problem. If legislators know they cannot influence the district lines, there is little reason to fight over them. The challenge becomes agreeing on the algorithm beforehand.
One could imagine a hierarchy such as:
Equal population.
Contiguity.
Maximize compactness (perhaps measured mathematically).
Generate districts automatically from census geography.
No access to party registration, voting history, incumbency, or election results.
The algorithm would then produce the same map no matter which party controlled the legislature.
Of course, arguments would shift to the choice of compactness measure, handling of census blocks, treatment of islands, and so forth. But those disputes tend to be narrower and more transparent than disputes over thousands of hand-drawn boundary decisions.
There's an old observation in political science: gerrymandering is possible because district boundaries are discretionary. Every reform that reduces discretion tends to reduce opportunities for gerrymandering. Convexity, compactness rules, independent commissions, and computer-generated maps are all different ways of attacking the same underlying issue.
Whether a fully automated system is best is ultimately a political judgment, but from a systems-design perspective it has an appealing quality: it treats districting as a technical optimization problem rather than a prize to be captured by whichever party happens to be in power after the census. That is very much in the spirit of how engineers and actuaries often approach fairness problems—define the rules in advance, then let the procedure run without regard to who benefits in a particular cycle.
Then there is the problem of the population data. An ongoing census rather than decal seems preferable, again by another, separate independent board. That would mean annual or even monthly adjustments, with a rule to prefer adjustments that leave sitting representatives in the same district.
You're describing a system that pushes the logic of depoliticization further: not only should map-drawing be independent, but the underlying population data should be continuously updated and independently maintained.
There are some attractive features to that idea:
District populations would remain much closer to equal over time.
Fast-growing and declining regions would be reflected sooner rather than waiting for the next census cycle.
The incentive to fight over a once-per-decade census count would be reduced.
Small annual adjustments might be less disruptive than large decennial redrawings.
On the other hand, continuous adjustment introduces new complications.
Representation stability. If district boundaries changed every month, or even every year, voters might find themselves repeatedly switching districts and representatives. Representatives would also face changing constituencies during their terms.
Administrative complexity. Election officials, parties, candidates, and voters would all need to know which district applies at any given time. Frequent changes increase the chance of confusion.
Data quality. A decennial census is expensive partly because it attempts a very thorough count. More frequent updates would likely rely on estimates, administrative records, or sampling methods. Those may be more current but can introduce different kinds of error.
Your suggestion of preferring adjustments that keep incumbents in the same district is interesting because it recognizes a practical concern. However, some reformers would object that incumbency should play no role whatsoever in district design, even as a tie-breaker. Others might argue that maintaining continuity of representation has value independent of the particular officeholder.
One possibility is a rule aimed at constituent stability rather than incumbent stability. For example:
Among maps that satisfy the population requirements, choose the one that minimizes the number of people moved into a different district.
That focuses on voters rather than representatives. It tries to preserve continuity without explicitly protecting incumbents.
In a broader sense, your proposal highlights a tension that appears in many systems. There are two competing goals:
Accuracy — districts should reflect current population as closely as possible.
Stability — voters and representatives should know where they stand and not have the rules constantly changing.
Different countries and reform proposals place those goals at different points on the spectrum. A continuously updated system moves toward accuracy; a once-per-decade system moves toward stability. The question is less whether one is right and the other wrong than where the balance should be struck.
I agree, the representative can still serve out their term and either run for the seat in the new district or move. I think constantly updating population data would serve many useful purposes for both public and private interests and therefor useful on its own.
That's a coherent way to separate the issues.
If the goal is fair representation, then the representative's residence becomes almost incidental. The district exists to represent the people living in it, not to preserve the career of the current officeholder. Under that view, if a boundary adjustment leaves Representative Smith outside the district they currently represent, Smith simply finishes the term and then decides whether to run in the new district, move, retire, or seek another office.
In fact, in the U.S. Constitution, members of the House must live in the state they represent, but not necessarily in the district itself. So historically there has been less emphasis on district residency than many people assume.
As for a continuous population register, there are indeed benefits beyond redistricting.
Governments use population data for:
Infrastructure planning.
School construction and staffing.
Transportation.
Public health.
Emergency management.
Pension and social service forecasting.
Tax revenue projections.
Private organizations use similar data for:
Retail location decisions.
Insurance risk analysis.
Housing development.
Utilities planning.
Telecommunications networks.
From a systems perspective, a decennial census is a bit of an oddity. It originated in an era when gathering population information was slow, expensive, and largely manual. Today, many countries maintain ongoing population registers or continuously updated administrative databases and therefore do not rely on a massive count every ten years.
The tradeoff is usually privacy and governance. A continuously updated population system requires government to maintain more current information about where people live and often to integrate records from tax authorities, immigration agencies, vital records, and other databases. Some societies are comfortable with that; others are wary of the concentration of information.
Your instinct that the data would have value independent of redistricting is hard to dispute. If a country could obtain accurate, timely population information in a way citizens trusted, many public and private decisions could be made on a sounder factual basis. Redistricting would merely be one application among many.
There's a certain engineering elegance to the architecture you're sketching:
An independent population agency continuously maintains population data.
An independent algorithm generates districts from that data according to fixed rules.
Elected officials have no role in either process.
The more those functions are separated, the fewer opportunities there are for political actors to influence the inputs or the outputs. Whether a society would accept such a technocratic arrangement is a political question, but as a system design it has a clear internal logic.