Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  show that the union of translates of a convex body in three dimensional space can have a cubic number holes in the worst case, where a hole in a set is a connected component of its compliment. This refutes a 20-year-old conjecture. As a consequence,  also obtain improved lor bounds on the complexity of motion planning problems and of Voronoi diagrams with convex distance functions.
['Boris Aronov', 'Otfried Cheong', 'Michael Gene Dobbins', 'Xavier Goaoc']
I am writing a research paper on a idea The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions
[{"from": "human", "value": "I am writing a research paper on a idea The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  show that the union of translates of a convex body in three dimensional space can have a cubic number holes in the worst case, where a hole in a set is a connected component of its compliment. This refutes a 20-year-old conjecture. As a consequence,  also obtain improved lor bounds on the complexity of motion planning problems and of Voronoi diagrams with convex distance functions."}]