I am kind of TAing for a class of real analysis, and I would like to speak a little about convex sets tomorrow, and explain why they are important. What kind of examples could I give? I was thinking of speaking a little about convex optimization, but as it is really out of my field of knowledge I can only give a very superficial overview of the subject.

Another possibility would be speaking of the Hahn-Banach separation theorem and of Krein-Milman's theorem, but I'm afraid it would be too complicated for them, and maybe not so interesting.

Do anyone have any good alternative ideas?

  • $\begingroup$ Probability distributions form a convex space, where convex combination means randomly choosing one of several random processes. $\endgroup$ – celtschk Apr 13 '14 at 12:25
  • $\begingroup$ @celtschk Thanks very much, but they haven't had any course in probability theory yet, so I would avoid that example. $\endgroup$ – Daniel Robert-Nicoud Apr 13 '14 at 12:38
  • $\begingroup$ The classic horseshoe example might serve as a good introductory instance of set that is not convex. After defining a convex set, another good example would be to ask them what is the smallest convex set (convex hull) with 3 points. A nice theorem where convexity plays a role is the Gauss-Lucas theorem which shows up in complex analysis. $\endgroup$ – Mustafa Said Apr 13 '14 at 12:38
  • $\begingroup$ @MustafaSaid Thanks, I didn't know about Gauss-Lucas' theorem. It is a nice example and I will probably include it. If you come up with anything else let me know. $\endgroup$ – Daniel Robert-Nicoud Apr 13 '14 at 12:41

Use Bezier curves. Since the Bernstein polynomials $\phi^m_i$ form a partition of unity, the Bezier curve $$ \mathbf{C}(t) = \sum _{i=0}^m \phi^m_i \mathbf{P}_i $$ lies inside the convex hull of the points $\mathbf{P}_0, \ldots, \mathbf{P}_m$. You can often do (inexpensive) computations on the convex hull in order to avoid (expensive) computations on the curve itself. For example, if you need to find intersections of two Bezier curves, you can quit if you find that their convex hulls don't intersect. And, if their convex hulls do intersect, you can subdivide, etc, etc.


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