See if you can show that the circumcircle of a bounded planar convex set K is unique.
Show that K intercepts the circumcircle either in two diametrically opposite points, or in three points, not all on the same semicircle.
2.
The incircle of a bounded planar convex set is unique.
True ; False
False. Consider a non-square rectangle: this has many congruent incircles.
3.
In Bender’s Theorem, why have we used < and not ?
The limiting case is for the (semi-) infinite strip, but now neither A nor P are defined, being infinite.
4.
Since we have used ‘ < 1/2’ in Bender’s Theorem, this means that the bound can be improved.
True ; False
False. We are touching on the idea of supremum here. For any number e, no matter how small, we can find a long rectangle of unit width for which A/P > 1/2 – e. Thus the bound cannot be improved.
5.
Look up Steiner’s ‘proof’ of the isoperimetric inequality. Do you understand the text comment about existence?
Here is an existence example. Thm: The smallest positive real number q is 1. Proof: If q < 1, then q2 < q, and q is not smallest. In fact, there is no smallest positive real number.
6.
Show that the inequality (w – 1)(d – 1) 1 can be rewritten involving the reciprocals of w, d.
(w – 1)(d – 1) 1 is equivalent to wd – w – d + 1 1, or w + d wd. That is, 1/d + 1/w 1.
?
1. Proof: If q < 1, then q2 < q, and q is not smallest. In fact, there is no smallest positive real number.