Jacob Siehler, a friend from Mathstodon, and Assistant Professor at Gustavus Adolphus College, offered several good topics for the letter ‘C’. I picked the one that seemed to connect to the greatest number of other topics I’ve covered recently.
It’s easy to say what convex is, if we’re talking about shapes in ordinary space. A convex shape is one where the line connecting any two points inside the shape always stays inside the shape. Circles are convex. Triangles and rectangles too. Star shapes are not. Is a torus? That depends. If it’s a doughnut shape sitting in some bigger space, then it’s not convex. If the doughnut shape is all the space there is to consider, then it is. There’s a parallel here to prime numbers. Whether 5 is a prime depends on whether you think 5 is an integer, a real number, or a complex number.
Still, this seems easy to the point of boring. So how does Wolfram Mathworld match 337 items for ‘convex’? For a sense of scale, it has only 112 matches for ‘quadrilateral’. This is a word used almost as much as ‘quadrilateral’, with 370 items. Why?
Why is that it’s one of those terms that sneaks in everywhere. Some of it is obvious. There’s a concept called “star-convex”, where two points only need a connection by some path. It doesn’t have to be a straight line. That’s a familiar mathematical trick, coming up with a less-demanding version of a property. There’s the “convex hull”, which is the smallest convex set that contains a given set of points. We even come up with “convex functions”, functions of real numbers. A function’s convex if, the space above the graph of a function is convex. This seems like stretching the idea of convexity rather a bit.
Still, we wouldn’t coin such a term if we couldn’t use it. Well, if someone couldn’t use it. The saving thing here is the idea of “space”. We get it from our idea of what space is from looking around rooms and walking around hills and stuff. But what makes something a space? When we look at what’s essential? What we need is traits like, there are things. We can measure how far apart things are. We have some idea of paths between things. That’s not asking a lot.
So many things become spaces. And so convexity sneaks in everywhere. A convex function has nice properties if you’re looking for minimums. Or maximums; that’s as easy to do. And we look for minimums a lot. A large, practical set of mathematics is the search for optimum values, the set of values that maximize, or minimize, something. You may protest that not everything we’re intersted in is a convex function. This is true. But a lot of what we are interested in is, or is approximately.
This gets into some surprising corners. Economics, for example. The mathematics of economics is often interested in how much of a thing you can make. But you have to put things in to make it. You expect, at least once the system is set up, that if you halve the components you put in you get half the thing out. Or double the components in and get double the thing out. But you can run out of the components. Or related stuff, like, floor space to store partly-complete product. Or transport available to send this stuff to the customer. Or time to get things finished. For our needs these are all “things you can run out of”.
And so we have a problem of linear programming. We have something or other we want to optimize. Call it . It depends on a whole range of variables, which we describe as a vector . And we have constraints. Each of these is an inequality; we can represent that as demanding some functions of these variables be at most some numbers. We can bundle those functions together as a matrix called . We can bundle those maximum numbers together as a vector called . So the problem is finding . Also, we demand that none of these values be smaller than some minimum we might as well call 0. The range of all the possible values of these variables is a space. These constraints chop up that space, into a shape. Into a convex shape, of course, or this paragraph wouldn’t belong in this essay. If you need to be convinced of this, imagine taking a wedge of cheese and hacking away slices all the way through it. How do you cut a cave or a tunnel in it?
So take this convex shape, called a polytope. That’s what we call a polygon or polyhedron if we don’t want to commit to any particular number of dimensions of space. (If we’re being careful. My suspicion is ‘polyhedron’ is more often said.) This makes a shape. Some point in that shape has the best possible value of . (Also the worst, if that’s your thing.) Where is it? There is an answer, and it gives a pretext to share a fun story. The answer is that it’s on the outside, on one of the faces of the polytope. And you can find it following along the edges of those polytopes. This we know as the simplex method, or Dantzig’s Simplex Method if we must be more particular, for George Dantzig. Its success relies on looking at convex functions in convex spaces and how much this simplifies finding things.
Usually. The simplex method is one of polynomial-order complexity for normal, typical problems. That’s a measure of how much longer it takes to find an answer as you get more variables, more constraints, more work. Polynomial is okay, growing about the way it takes longer to multiply when you have more digits in the numbers. But there’s a worst case, in which the complexity grows exponentially. We shy away from exponential-complexity because … you know, exponentials grow fast, given a chance. What saves us is that that’s a worst case, not a typical case. The convexity lets us set up our problem and, rather often, solve it well enough.
Now the story, a mutation of which it’s likely you encountered. George Dantzig, as a student in Jerzy Neyman’s statistics class, arrived late one day to find a couple problems on the board. He took these to be homework, and struggled with the harder-than-usual set. But turned them in, apologizing for them being late. Neyman accepted the work, and eventually got around to looking at it. This wasn’t the homework. This was some unsolved problems in statistics. Six weeks later Neyman had prepared them for publication. A year later, Neyman explained to Dantzig that all he needed to earn his PhD was put these two papers together in a nice binder.
This cute story somehow escaped into the wild. It became an inspirational tale for more than mathematics grad students. That part’s easy to see; it has most everything inspiration needs. It mutated further, into the movie Good Will Hunting. I do not know that the unsolved problems, work done in the late 1930s, related to Dantzig’s simplex method, proved after World War II. It may be that they are simply connected in their originator. But perhaps it is more than I realize now.
I hope to finish off the word ‘Mathematics’ with the letter S next week. This week’s essay, and all the essays for the Little Mathematics A-to-Z, should be at this link. And all of this year’s essays, and all the A-to-Z essays from past years, should be at this link. Thank you for reading.