Let P be a set of n points in the plane, no three collinear. A convex polygon of P is called empty if no point of P lies in its interior. An empty partition of P is a partition of P into empty convex polygons. Let k be a positive integer and N π k (P) be the number of empty convex k-gons in an empty partition π of P. Define g k (P) =: max{N π k (P) : π is an empty partition of P}, G k (n) =: min{g k (P) : |P| = n}. We mainly study the case of k = 4 and get the result that G 4 (n) ≥ 9n 38 . For specified n = 21 × 2 k−1 − 4 (k ≥ 1), we obtain the better bound G 4 (n) ≥ 5n−1 21 .
Flocculating agents are widely used in various industrial fields including wastewater treatment, tap water production, dredging/downstream processing, food and fermentation processes 1. A variety of flocculants, such as inorganic aluminum or ferric salts and organic synthetic high polymers, have been widely used due to their effective flocculating activity and low cost. However, they inherit the drawback of being less biodegradable and producing carcinogenic monomers during degradation 2. In recent years, microbial-produced bioflocculants have received increasingly scientific and biotechnological attention due to their nontoxic, biodegradability and lack of secondary pollution of their degradation intermediates 3-5. Bioflocculants are essentially polymers produced by microorganisms during growth, with flocculating activities that are dependent on the characteristics of the flocculants. Over the past decades, some microorganisms, including algae, bacteria, actinomyces and fungi, have been reported to produce bioflocculants 2,6-9. The composition of bioflocculants has been reported to include polysaccharides, protein, nucleic acid or PHB 7,10-12. Low flocculating capability and large dosage requirement, however, have been major problems in bioflocculant development for practical application 13. Consequently, many researchers pay attention to discover novel efficient bioflocculants from microorganisms in varied environments and study their flocculating mechanisms over the past decades 14-16. We have isolated several flocculant-producing strains from activated sludge of wastewater treatment plant. Among them,
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