The midterm had 6 questions: 3 proofs done in class (Cauchy-Schwarz, orthogonal projection theorem, and parallelotope volume theorem — these can all be found in my notes), a question about simplex centroids, a question about a “sum-of-squares”-minimizing point in the plane, and a question about angles between orthogonal complements (which no one claimed to have solved).
I’ll post the questions (#4 to #6, since as I said the rest were just proofs from class) here. Feel free to post your solutions, or any ideas. If I have time, I’ll post my solution to #4 below (although it took me a lot of algebra to prove uniqueness…)
- [#4 on exam] Let be an -simplex with centroid . For each with , define to be the centroid of the -simplex . Define also to be the line passing through and . Show that the lines all intersect uniquely at the point .
- [#5 on exam] Given , find a point minimizing the sum . Prove this point is unique.
- [#6 on exam] Let be hyperspaces in . Prove that .
Also, sorry about the horribly outdated Spherical Geometry section in my notes. There are so many diagrams in my notebook that I haven’t had the time to transplant into LaTeX. I’ll do it the minute I get some spare time. Alternatively, you can just look at Professor New’s notes…