## Convex Combinations

Jack Sprat could eat no fat; his wife could eat no lean.  And so, between the two of them, they licked the platter clean.

With my girlfriend and I the meals are not divided (100%,0) or (0,100%).  But the same concept applies:  I’ll have 25% of her beer and she’ll have 25% of mine.  The nursery rhyme stands in for the general idea of a general convex combination — any such combination as (53%, 47%), (1%, 99%), or (25%, 75%).

That’s what a convex combination is.

It’s written with a λ and looks so much more mystifying that way:

$\dpi{300} \bg_white \lambda \mathbf{A} + (1-\lambda) \mathbf{B}, \quad \quad \lambda \in [0,1] \,$

But just say that A and B are two things, like in the case above, two 4-D vectors each containing (amount of Guinness I have;   amount of Guinness she has;   amount of Old Rasputin I have;   amount of Old Rasputin she has).  The quantity

$\dpi{300} \bg_white \lambda \mathbf{A} + (1-\lambda) \mathbf{B}$

mustn’t total up to more beers than we bought … which is common sense, really.

## Wax Philosophical

So if the definition makes sense, let me just throw out a few mind-expanding ideas you can conceive with it:

• Mixing colours is a convex combination. (R, G, B) is a linear 3-space. So is `(H, S, V)` — and too, there is a reversible transformation from one to the other. `(C,M,Y,K)` is a 4-space so the transformation can’t be so simple.
• Can you then say that one colour is “between” two others?
• Can you imagine a colour that’s a convex combination of three colours? Would that make sense?

• On from colours to ideas. Have you ever noticed that if people are taught two competing theories in a class, then they try to balance between them? I noticed this in political theory, anthropology, and philosophy classes.
• The Economist's Which MBA ranking allows you to adjust the importance of various factors. Typical college ranking systems do the following: (1) observe and score schools on several facts, (2) combine these (independent or not) dimensions into a total order (3) using the weighted-average method. The weightings are arbitrary, which mean the ranking would be different for someone with different priorities. If I assume your preference weighting is linear then .
• I have a pet theory that it’s very natural for people to want to compromise among the ideas that they’re given — i.e., occupy some convex combination rather than a "corner".
• My pet theory goes further to say that revolutionary ideas don’t necessarily have to be “orthogonal” — don’t have to be completely radical and unintelligible according to current ideas — to permit novel thought.

If the idea has even just a little bit of a unique notion (points just a wee bit into a new dimension), then that idea can be combined, linearly, with old ideas, and the entire dimension of new ideas is opened up.

• Lastly, science. You can have a convex combination of quantum states. That’s where the concept of superposition comes from.

From a geometrical perspective, convex combinations happen on the surface of a high-dimensional sphere, restricted to the positive “octant” i.e. all angles between `[0°,90°]`.

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1. isomorphismes posted this