A better decomposition theorem for simply connected m-convex sets |
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Authors: | Marilyn Breen |
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Affiliation: | The university of Oklahoma, Norman, OK 73019, U.S.A. |
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Abstract: | A set S in Rd is said to be m-convex, m ? 2, if and only if for every m points in S, at least one of the line segments determined by these points lies in S. For S a closed m-convex set in R2, various decomposition theorems have been obtained to express S as a finite union of convex sets. However, the previous bounds may be lowered further, and we have the following result:In case S is simply connected, then S is a union of σ(m) or fewer convex sets, where .Moreover, this result induces an improved decomposition in the general case as well. |
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