## Convexity properties of the cone of nonnegative polynomials

Venue: | Discrete Comput. Geom |

Citations: | 8 - 3 self |

### BibTeX

@ARTICLE{Blekherman_convexityproperties,

author = {Grigoriy Blekherman},

title = {Convexity properties of the cone of nonnegative polynomials},

journal = {Discrete Comput. Geom},

year = {},

volume = {32},

pages = {345--371}

}

### OpenURL

### Abstract

We study metric properties of the cone of homogeneous non-negative multivariate polynomials and the cone of sums of powers of linear forms, and the relationship between the two cones. We compute the maximum volume ellipsoid of the natural base of the cone of nonnegative polynomials and the minimum volume ellipsoid of the natural base of the cone of powers of linear forms and compute the coefficients of symmetry of the bases. The multiplication by (x 2 1 +... + x2 n) m induces an isometric embedding of the space of polynomials of degree 2k into the space of polynomials of degree 2(k + m), which allows us to compare the cone of non-negative polynomials of degree 2k and the cone of sums of 2(k + m)-powers of linear forms. We estimate the volume ratio of the bases of the two cones and the rate at which it approaches 1 as m grows.

### Citations

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Citation Context ...1 . The family of polynomials Qn,d(t) are known as the Legendre polynomials and are special cases of ultraspherical (or Gegenbauer) polynomials. For many identities satisfied by these polynomials see =-=[11]-=- and [12]. 9s2.2 Loewner and John Ellipsoids Let K be a convex body in a finite dimensional real vector space V . There exists a unique ellipsoid of maximal volume contained in K, known as John’s elli... |

307 |
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(Show Context)
Citation Context ...s f on the sphere S n−1 . We denote the image subspace of Pn,d by H ∗ n,d−2l : H ∗ n,d−2l = {f ∈ Pn,d | f = r 2l g for some g ∈ Hn,d−2l} We need some facts about the representations φn,d, see [6] and =-=[12]-=-. Theorem 2.1 Hn,d is an irreducible SO(n)-module, and, therefore, H ∗ n,d is an irreducible submodule of Pn,d. Furthermore, Pn,d splits into irreducible submodules as follows: Pn,d = ⌊d/2⌋ � i=0 r 2i... |

233 |
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(Show Context)
Citation Context ... R n . For the proof and discussion see [1]. There also exists a unique ellipsoid of minimal volume containing K, known as the Loewner ellipsoid of K; we will denote it by LK. It was shown by John in =-=[5]-=- that if B n contains K, then the same condition on points in the intersection of boundaries is necessary and sufficient for a unit ball B n to be the Loewner ellipsoid of K. Definition. For a convex ... |

215 |
algebraic geometry
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(Show Context)
Citation Context ... cases when Cn,2k = Sqn,2k,[4]. Hilbert’s 17th problem, solved in affirmative by Artin and Schreier in the 1920’s, asked whether every nonnegative polynomial is a sum of squares of rational functions =-=[3]-=-. Constructive aspects of Hilbert’s problem still draw attention today [3],[7]. For a discussion of some algebraic properties of the cone of sums of powers of linear forms we refer to [8]. To our know... |

106 | An elementary introduction to modern convex geometry
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(Show Context)
Citation Context ...compact convex body K contains a unique ellipsoid of maximum volume, known as John’s ellipsoid of K. Also, K is contained in a unique ellipsoid of minimum volume, known as the Loewner ellipsoid of K, =-=[1]-=-. A crude, yet interesting, measure of symmetry of K is its coefficient of symmetry about a point v in the interior of K. The coefficient of symmetry of K about v is defined as the largest α ∈ R such ... |

98 | Some concrete aspects of Hilbert’s 17th problem
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(Show Context)
Citation Context ...by Artin and Schreier in the 1920’s, asked whether every nonnegative polynomial is a sum of squares of rational functions [3]. Constructive aspects of Hilbert’s problem still draw attention today [3],=-=[7]-=-. For a discussion of some algebraic properties of the cone of sums of powers of linear forms we refer to [8]. To our knowledge, however, these cones have not been studied as general convex objects, p... |

66 |
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(Show Context)
Citation Context ... = f ∈ Pn,2k ⏐ � ⏐ f = l 2k i for some linear forms � li ∈ Pn,1 . i i The study of algebraic properties of these cones goes back to Hilbert, who described explicitly all the cases when Cn,2k = Sqn,2k,=-=[4]-=-. Hilbert’s 17th problem, solved in affirmative by Artin and Schreier in the 1920’s, asked whether every nonnegative polynomial is a sum of squares of rational functions [3]. Constructive aspects of H... |

56 |
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(Show Context)
Citation Context ...al functions [3]. Constructive aspects of Hilbert’s problem still draw attention today [3],[7]. For a discussion of some algebraic properties of the cone of sums of powers of linear forms we refer to =-=[8]-=-. To our knowledge, however, these cones have not been studied as general convex objects, possessing invariants based on convexity. In this paper we look at some convex properties of these cones. Let ... |

47 |
Analysis of Spherical Symmetries in Euclidean Spaces
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(Show Context)
Citation Context ...nction as f on the sphere S n−1 . We denote the image subspace of Pn,d by H ∗ n,d−2l : H ∗ n,d−2l = {f ∈ Pn,d | f = r 2l g for some g ∈ Hn,d−2l} We need some facts about the representations φn,d, see =-=[6]-=- and [12]. Theorem 2.1 Hn,d is an irreducible SO(n)-module, and, therefore, H ∗ n,d is an irreducible submodule of Pn,d. Furthermore, Pn,d splits into irreducible submodules as follows: Pn,d = ⌊d/2⌋ �... |

25 |
Representations of Finite and Compact Groups, Graduate
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Citation Context ...obability measure µ on G, called the Haar measure. From existence of Haar measure it easily follows that there exists a G-invariant scalar product 〈 , 〉 that makes φ into an orthogonal representation =-=[9]-=-. We will use the shorthand notation of g(v) to denote the action of φ(g) on a vector v ∈ V . Let v ∈ V and let Ov be the orbit of v, Ov = � g(v) | g ∈ G � . Let W denote the affine span of Ov, � � ⏐ ... |

20 |
Oscillatory integrals and spherical harmonics
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(Show Context)
Citation Context ... Corollary 6.4) Estimates (1) and (2) above are sharp and we also provide all extreme polynomials for them. For a different proof of (4) by Barvinok and a discussion of applications see [2]. Sogge in =-=[10]-=-, and Duoandikoetxea in [4] derive some related interesting inequalities. The rest of the article is structured as follows: Section 2 contains the known results necessary for the rest of the paper. In... |

9 |
Estimating L ∞ norms by L 2k norms for functions on orbits
- Barvinok
(Show Context)
Citation Context ...ntegers l. (cf Corollary 6.4) Estimates (1) and (2) above are sharp and we also provide all extreme polynomials for them. For a different proof of (4) by Barvinok and a discussion of applications see =-=[2]-=-. Sogge in [10], and Duoandikoetxea in [4] derive some related interesting inequalities. The rest of the article is structured as follows: Section 2 contains the known results necessary for the rest o... |

8 |
Reverse Hölder inequalities for spherical harmonics
- Duoandikoetxea
- 1987
(Show Context)
Citation Context ... = f ∈ Pn,2k ⏐ � ⏐ f = l 2k i for some linear forms � li ∈ Pn,1 . i i The study of algebraic properties of these cones goes back to Hilbert, who described explicitly all the cases when Cn,2k = Sqn,2k,=-=[4]-=-. Hilbert’s 17th problem, solved in affirmative by Artin and Schreier in the 1920’s, asked whether every nonnegative polynomial is a sum of squares of rational functions [3]. Constructive aspects of H... |

1 |
Reverse Hölder inequalities for spherical harmonics
- Duoandkoetxea
- 1987
(Show Context)
Citation Context ...{ ∑ Lf = f ∈ Pn,2k ⏐ f = l 2k } i for some linear forms li ∈ Pn,1 . i The study of algebraic properties of these cones goes back to Hilbert, who described explicitly all the cases when Cn,2k = Sqn,2k,=-=[4]-=-. Hilbert’s 17th problem, solved in affirmative by Artin and Schreier in the 1920’s, asked whether every nonnegative polynomial is a sum of squares of rational functions [3]. Constructive aspects of H... |