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:

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

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

— and too, there is a reversible transformation from one to the other.**(**H**,**S**,**V**)**

is a 4-space so the transformation can’t be so simple.**(**C**,**M**,**Y**,**K**)** - 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°]`

.