By R. Miller, L. Boxer

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Aurenhammer, and H. Krasser Proof. We ﬁrst partition P into strongly pseudo-convex polygons. Diagonals on non-convex geodesics and corner tangents are used, such that each introduced diagonal is incident to some corner in both polygons it bounds. ) Each strongly pseudoconvex polygon Q with more than 3 corners is partitioned further as follows. Integrate a nice pseudo-triangle ∇ with small cut for Q, whose existence is guaranteed by Lemma 5. Because ∇ is nice, it does not violate the pointedness of any vertex.

If σ1 < σ2 , then the radius of the circumcircle of σ2 is larger than the radius of the circumcircle of σ1 . So, in a chain of triangles related by < relation the circumradii of the triangles can never decrease, thus making it impossible for <∗ to be cyclic. This means that, for each triangle σ , there is a triangle σ containing a maximum x such that σ <∗ σ. We will say that σ ﬂows into σ. The following lemma holds in R2 . Lemma 4. Let σ be a triangle containing a maximum x. We have F (x) = σ <∗ σ σ.

We conclude that there are at least 2 i + 3 sparse faces in total. So the mean number of sparse faces per point in M exceeds two, which implies that there exists a point p ∈ M incident to three sparse faces. Among them, let f be the face that contains the most points, which are at least 3i . We take the two geodesics that span f to partition ∇. This yields two parts with at most 23i points each. Lemma 6 combines with Theorem 3 to the following partition theorem for augmented polygons. Theorem 4.

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