convexity

There was an econ textbook that gave a memorable example of a (qualified) exception to convexity: ice-cream and olives.

If they're polygonal, the polygon will not have perfectly flat faces due to finite precision and may have false convexity.

Measuring concavity/convexity of a generic 2D surface object doesn't seem to be a very common operation.

Turbulence depends on sail convexity, it can be changed too, not only sail angle!

So, the definition of convexity is exactly that statement, but where n equals two.

So with a few minor complications convexity generalises to Riemannian manifolds like the earth.

If it isn't, you walk along both directions along the (convex) polygon until you hit the first point that doesn't violate the convexity constraint.

Aside: even single-layer neural networks do not have convex error surfaces; convexity, and funky error surface geometry It's correct that single-layer neural nets are non-convex.

The problem with making risk aversion/convexity work with lotteries and such is that the price you're supposed to pay for lotteries is off your utility surface.

You are right about the non-convexity of the space, especially the weird interactions between the choice of which W_k to include and the shape of the manifold.

The general mathematical property of this asymmetry is convexity (which is explained in Figure 1); functions with larger gains than losses are nonlinear-convex and resemble financial options.

Consider another class, lets throw away convexity, consider functions that are non-differentiable and very rough locally but when filtered with a low pass filter is well behaved.

> "Topology ignores such things as corners, distances, and convexity." No, that's what topology is.

The hedge gives you protection against the price moves in the underlying and you effectively make money for small moves in the security, regardless of whether it goes up or down (due to "convexity").

See "Forward models: Supervised learning with a distal teacher" which trains two networks in parallel, one a forward model, one an inverse model, and uses the Jacobian to circumnavigate the problematic non-convexity of the inverse model.

Maths challenge: I wonder if you could prove this "probability of a random line segment violating convexity" definition equivalent to something given in terms of a ratio of different areas like the area to convex hull area suggestion below.