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2009

Convex Forms that are not Sums of Squares

Blekherman, Grigoriy

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An orbitope is the convex hull of an orbit of a point under the action of a compact group.

We derive bounds on volumes of sections of polar bodies of orbitopes, extending our previously developed methods.
As an application we realize the cone of convex forms as a section of the cone of nonnegative bi-homogeneous forms and estimate its volume.
A convex form has to be nonnegative, but it has not been previously shown that there exist convex forms that are not sums of squares.

Convex Forms that are not Sums of Squares · Around

Built on

J. Duoandikoetxea, Reverse Hölder Inequalities for Spherical Harmonics
1987
Earlier work this paper cites.
B. Reznick, Sums of even powers of real linear forms
Original
1992
Earlier work this paper cites.

Similar

J.H van Lint, Introduction to Coding Theory. Third Edition
1999
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A. Barvinok, Estimating L ∞ L^{\infty} norms by L 2 k L^{2k} norms for functions on orbits
2002
Cited alongside, same era.

Then

A. Barvinok, G. Blekherman, Convex Geometry of Orbits
2005
Later among the works it cites.
G. Blekherman, There are significantly more nonnegative polynomials than sums of squares
2006
Later among the works it cites.

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