This paper proposes an approach to extract skeletons from the character patterns. It first decomposes the pattern into a set of near-convex parts and then extracts skeletons from the parts. In shape decomposition stage, the convex hull information is used to identify the splitting paths. For the skeleton extraction, an operation that ties the adjacent strokes by a knot is developed. Our control...

Suppose d+ 1 absolutely continuous probability measures m0, . . . ,md on Rd are given. In this paper, we prove that there exists a point of Rd that belongs to the convex hull of d+1 points v0, . . . , vd with probability at least 2d (d+1)!(d+1) , where each point vi is sampled independently according to probability measure mi.

Let Xd,n be an n-element subset of {0, 1} chosen uniformly at random, and denote by Pd,n := conv Xd,n its convex hull. Let ∆d,n be the density of the graph of Pd,n (i.e., the number of one-dimensional faces of Pd,n divided by ` n 2 ́ ). Our main result is that, for any function n(d), the expected value of ∆d,n(d) converges (with d → ∞) to one if, for some arbitrary ε > 0, n(d) ≤ ( √ 2 − ε) holds...

We consider the following problem. Let n ≥ 2, b ≥ 1 and q ≥ 2 be integers. Let R and B be two disjoint sets of n red points and bn blue points in the plane, respectively, such that no three points of R∪B lie on the same line. Let n = n1 + n2 + · · · + nq be an integer-partition of n such that 1 ≤ ni for every 1 ≤ i ≤ q. Then we want to partition R∪B into q disjoint subsets P1 ∪P2 ∪ · · · ∪Pq th...

We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59 O(n−6) straight-edge triangulations of its convex hull. This improves the previous upper bound of 276.75n+O(log(n)).

For a point set P on the plane, a four element subset S ⊂ P is called a 4-hole of P if the convex hull of S is a quadrilateral and contains no point of P in its interior. Let R be a point set on the plane. We say that a point set B covers all the 4-holes of R if any 4-hole of R contains an element of B in its interior. We show that if |R| ≥ 2|B|+5 then B cannot cover all the 4-holes of R. A sim...

This paper presents a new variation of Tverberg’s theorem. Given a discrete set S of R, we study the number of points of S needed to guarantee the existence of an m-partition of the points such that the intersection of the m convex hulls of the parts contains at least k points of S. The proofs of the main results require new quantitative versions of Helly’s and Carathéodory’s theorems.