Convexity of indifference curve The convexity of an indifference curve results from the fact that the absolute value of its (negative) derivative, which is the marginal rate of substitution is decreasing. But why do we say that it's convex to the origin? 

What is an implicit function that is convex and concave to the origin?
I think what people mean when they say "convex to the origin" (or to any point $p$) is that the function is convex when looked at in a new basis, namely the basis resulting from a rotation such that the new x axis (call it x') is, up to a constant, tangent to the IC and the distance $|p-IC|$ is minimized by that point of tangency ($w$).