A pentagon and its type cone
- Author:
- Bennet Goeckner
The gold cone on the right is the "type cone" of the pentagon on the left. The purple point on the right corresponds to the current realization of the pentagon. (The type cone of a polytope P is the space of all weak Minkowski summands of P.)
Changing the heights of the hyperplanes on the left (by changing a, d, and f) will move the purple point. The point will stay in the type cone as long as all five hyperplanes touch the pentagon.
Points on the boundary of the type cone correspond to degenerate pentagons (i.e., ones that have fewer than 5 edges). Extremal rays on the type cone correspond to deformations that have no nontrivial Minkowski summands.
Note: This type cone lives in R^5 but is three-dimensional. I have fixed b and c for this realization. (This does not change the type cone here in any fundamental way.)