# Approximating Nonnegative Polynomials via Spectral Sparsification

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*Lecture notes for Princeton ORFE course*, Lecture 15; available at https://www.princeton.edu/~amirali/Public/Teaching/ORF523.

*A bound for the number of vertices of a polytope with applications*, Combinatorica, 33 (2013), pp. 1--10.

*A Course in Convexity*, Grad. Stud. Math. 54, American Mathematical Society, Providence, RI, 2002.

*Convex geometry of orbits*, in Combinatorial and Computational Geometry, Math. Sci. Res. Inst. Publ. 52, Cambridge University Press, Cambridge, 2005, pp. 51--77.

*Approximating $L^{\infty}$ norms by $L^{2k}$ norms for functions on orbits*, Found. Comput. Math., 2 (2002), pp. 393--412.

*Thrifty approximations of convex bodies by polytopes*, Int. Math. Res. Not., 2014 (2014), pp. 4341--4356.

*Deciding polyhedrality of spectrahedra*, SIAM J. Optim., 25 (2015), pp. 1873--1884.

*Convexity properties of the cone of nonnegative polynomials*, Discrete Comput. Geom., 32 (2004), pp. 345--371.

*Semidefinite Optimization and Convex Algebraic Geometry*, MOS-SIAM Ser. Optim. 13, SIAM, Philadelphia, PA, 2013.

*Gaussian Measures*, Math. Surveys Monogr. 62, American Mathematical Society, Providence, RI, 1998.

*Remarks on the growth of Lp-norms of polynomials*, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1745, Springer, Berlin, 2000.

*The matching polytope does not admit fully-polynomial size relaxation schemes*, in Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 2015, pp. 837--846.

*Approximation limits of linear programs (beyond hierarchies)*, Math. Oper. Res., 40 (2015), pp. 756--772.

*Geometry of Isotropic Convex Bodies*, Math. Surveys Monogr. 196, American Mathematical Society, Providence, RI, 2014.

*Approximating convex bodies and matrices through Kadison--Singer*, Int. Math. Res. Not. (2017), https://doi.org/10.1093/imrn/rnx206.

*Minimal ellipsoids and their duals*, Rend. Circ. Mat. Palermo (2), 37 (1988), pp. 35--64.

*Even symmetric sextics*, Math. Z., 195 (1987), pp. 559--580.

*Lower bounds for polynomials with simplex Newton polytopes based on geometric programming*, SIAM J. Optim., 26 (2016), pp. 1128--1146.

*Extremum problems with inequalities as subsidiary conditions*, in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, New York, NY, 1948, pp. 187--204.

*Almost tight bounds for $\varepsilon$-nets*, Discrete Comput. Geom., 7 (1992), pp. 163--173.

*Interlacing families I: Bipartite Ramanujan graphs of all degrees*, Ann. Math., 182 (2015), pp. 307--325.

*Interlacing families II: Mixed characteristic polynomials and the Kadison--Singer problem*, Ann. Math., 182 (2015), pp. 327--350.

*Lectures on Discrete Geometry*, Grad. Texts in Math. 212, Springer, New York, 2002.

*Approximating a Convex Body with a Polytope Using the Epsilon-Net Theorem*, preprint, https://arxiv.org/pdf/1705.07754.pdf, 2017.

*Exploiting Algebraic Structure in Sum of Squares Programs*, in Positive Polynomials in Control, D. Henrion and A. Garulli, eds., Lect. Notes Control Inf. Sci. 312, Springer, Berlin, 2005, pp. 181--194.

*On polynomials in normally distributed random variables*, Probab. Theory Appl., 37 (1992), pp. 692--694.

*Optimizing $n$-variate $(n+k)$-nomials for small $k$*, Theoret. Comput. Sci., 412 (2011), pp. 1457--1469.

*Polyhedra, spectrahedra, and semidefinite programming*, in Topics in Semidefinite and Interior-Point Methods, Fields Inst. Commun. 18, American Mathematical Society, Providence, RI, 1998, pp. 27--38.

*Random vectors in the isotropic position*, J. Funct. Anal., 164 (1999), pp. 60--72.

*Contact points of convex bodies*, Israel J. Math., 101 (1997), pp. 93--124.

*Spectral Sparsification and Restricted Invertibility*, Ph.D. Dissertation, Yale University, 2010; available at https://math.berkeley.edu/~nikhil/dissertation.pdf.

*Minimum-Volume Ellipsoids: Theory and Algorithms*, MOS-SIAM Ser. Optim. 23, SIAM, Philadelphia, PA, 2016.

*High-Dimensional Probability: An Introduction with Applications in Data Sciences*, Camb. Ser. Stat. Probab. Math. 47, Cambridge University Press, Cambridge, 2018.

*Introduction to the non-asymptotic analysis of random matrices*, in Compressed Sensing: Theory and Applications, Cambridge University Press, Cambridge, 2012, pp. 210--268.

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**Submitted**: 20 March 2017

**Accepted**: 8 January 2019

**Published online**: 21 March 2019

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