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Bachir EL KHADIR(bkhadirprinceton.edu) Abstract: A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, Blekherman answered the question in the negative by showing through volume arguments that for high enough number of variables, there must be convex forms of degree as low as 4 that are not sums of squares. Remarkably, no examples are known to date. In this paper, we show that all convex forms in 4 variables and of degree 4 are sums of squares. We also show that if a conjecture of Blekherman related to the socalled CayleyBacharach relations is true, then the same statement holds for convex forms in 3 variables and of degree 6. These are the two minimal cases where one would have any hope of seeing convex forms that are not sums of squares (due to known obstructions). A main ingredient of the proof is the derivation of certain "generalized CauchySchwarz inequalities" which could be of independent interest. Keywords: Convex Polynomials, Sum of Squares of Polynomials, CauchySchwarz Inequality Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Category 2: Global Optimization (Theory ) Category 3: Linear, Cone and Semidefinite Programming (Semidefinite Programming ) Citation: Download: [PDF] Entry Submitted: 09/22/2019 Modify/Update this entry  
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