Because we have videos and you can go back and forth a bit, I am going to do something a bit experimental: Rather than covering first convex sets, then convex/concave functions (link to 2018 notes) and then quasiconvex/quasiconcave functions (again link to 2018 notes), I will give videos that highlight the differences. You have already had the Hessians criteria for convexity/concavity, so I will present that next to the bordered Hessians criteria for quasiconvexity/quasiconcavity.

"Rough order":

Convex sets
Definitions and basics of convex functions and then "analogously" of quasiconvex functions
- ... and, f (quasi)concave iff -f (quasi)convex
Over to the "most familiar case": C².
- Already know: Hessian criteria for convexity/concavity
- Will be repeated and compared to: bordered Hessian criteria for quasiconvexity/quasiconcavity.
Throwing out the C². assumption, assuming only C¹ :
- A C¹ function is convex (resp. concave) iff first-order approximations never overestimate (resp. never underestimate); draw a tangent to x² and see that it is never above the graph. (This is something different than connecting two points on the graph; but, if you take a line through two points on the graph, move it vertically down until it is tangent.)
- The "similar" property for quasiconvexity/quasiconcavity: draw a level curve of a Cobb-Douglas. Draw a tangent to that level curve. The largest value of the Cobb-Douglas subject to being on that tangent line is ... the tangency point, not surprisingly. Quasiconcavity can be characterized by such a property.
Throwing out the C¹assumption, assuming only continuity:
- Example: take the function |t|. Consider any line through the origin and with slope in [-1,1]: that line is "as close as you get to" the tangency line of the C¹ case: it touches the graph, but is nowhere above it. It is called a subderivative ("subgradient" in dimension >1).
  - Convexity can be characterized by every point [in the interior of the domain] having a subgradient, a hyperplane that touches the graph at the point and is nowhere above the graph.
  - Turn it upside down for concavity: every point [in the interior of the domain] having a supergradient, a hyperplane that touches the graph at the point and is nowhere below the graph.
- For the quasiconcavity: Rather than your Cobb-Douglas level curve, draw a decreasing convex level curve with a cusp. Say, the level curve is not y=2/x , assume that y=max{3-x,2/x}. Now there will be several price vectors that lead you to adapt at (1,2), but whatever of those budget lines you draw, it is nowhere below the level curve.
  That property characterizes quasiconcavity. (Turn the argument upside down for quasiconvexity.)
Throwing out the continuity assumption?
- Some properties follow from the definitions with no additional assumptions. For example, the sum of concaves is concave ... but the sum of quasiconcaves need not be quasiconcave.
There is a particularly nice result about homogeneous positive quasiconcave (quasiconvex) functions, that gives criteria for when a quasiconcave is concave and a quasiconvex is convex. That can be used to decide the concavity of multivariate Cobb-Douglas functions by a shorter argument than by calculating minors of the the Hessian.
Concave and quasiconcave programs.
- Concave programs even without differentiability
- and, at the end of the course: precise necesssary Lagrange/Kuhn-Tucker conditions. With or without concavity. Turns out that concavity eliminates most of the ugly counterexamples that can break the Lagrange/Kuhn-Tucker methods.

First lecture "19"

Added after lectures: a page with "motivation".

Convex sets: youtube video 19a, https://youtu.be/uNghx92FqkU

Convex/convave functions: youtube video 19b, https://youtu.be/2lyamyJ9gN4

Quasiconvex/quasiconcave functions, you might have a look at this note about level sets. More examples can be found on the Wikipedia entry. Then youtube video 19c: https://youtu.be/Lnfw5Mhdu7g

Finally, convex preferences - and that is, strictly speaking, about convex sets. Still you probably appreciate it more (if at all) when you have gotten the connection to quasiconcavity: for a quasiconcave utility function and any reference consumption bundle x*, the set of everything (weakly) preferred to x* is convex. Now consider an Edgeworth box (like page 3 in this ECON4950 note); you may have observed that Edgeworth boxes are frequently drawn with the upper level sets convex. Stylized facts: for convex open sets (typically like the interior of such, that would be the set of everything you strictly prefer to x*), you can squeeze a hyperplane in between them. The separating hyperplane theorem (make that the "Hahn-Banach separation theorem" or "geometric Hahn-Banach theorem" if you are a mathematical economist[note below]) and is essentially the second theorem of welfare economics. A few words on that, 3 minutes: youtube video 19d, https://youtu.be/1Ueaa4Lf_9I

For those curious (ECON9xxx-curious!) about Hahn-Banach and the second theorem of welfare economics (read at own risk):

... it might show up in ECON9306 soon: one version is Theorem 75 in this manuscript by the Department's professor II Dirk Krueger. Various version of that manuscript have been curriculum at this department over the years. No, don't spend more than your five minutes do-something-else-while-practicing-social-distancing ... until after the exam.
When I got to that theorem - that must be my 150th math ECTS credit I think - we got a version that required a norm on the price space. That is kind of an unnatural restriction in economics, and I prefer the economists' more general version. (A "price vector" is something that prices a commodity vector in a linear way, and since you are only interested in price ratios, then requiring some measure of "magnitude" of a price vector is a bit nonsense. It so happens in Rⁿ that you have one, but beyond that ... why?!)