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MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves 1 Introduction 1 Related work 2 Preliminaries 2 Notation and definitions 2 Volume of the convex hull of a polynomial curve 3 Problems definition 3 Given PPn, find SSn 3 Given SSn, find PPn 3 Equivalent formulation 3 Constraints of Problems TeXFolio:prb11 and TeXFolio:prb22 3 Objective Function of 1 4 Objective Function of 2 4 Equivalent Formulation for Problems TeXFolio:prb11 and TeXFolio:prb22 4 Results 5 Results for n=1,2,,7 5 Results for n>7 9 MINVO basis applied to other curves 10 Polynomial curves of degree n, dimension k, and embedded in a subspace M of dimension m 10 Rational curves 12 Comparison with SLEFEs 12 Final remarks 13 Conversion between MINVO, Bernstein, and B-Spline 13 Tighter volumes for 1 via subdivision 13 When should each basis be used? 13 Conclusions and future work 14 Declaration of competing interest 14 Acknowledgments 14 Appendix A. Volume of the Convex Hull of a Polynomial Curve 15 Appendix B. Invertibility of the matrix A 15 Appendix C. Karush–Kuhn–Tucker conditions (for odd n) 15 Appendix D. Plot of a polynomial x(t)∈R [t] over t∈[-1,1] 16 Appendix E. Application of the MI

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