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We prove inequalities on symmetric tensor sums of positive definiteoperators. In particular, we prove multivariable operator inequalities inspiredby generalizations to the well-known Hlawka and Popoviciu inequalities. Ascorollaries, we obtain generalized Hlawka and Popoviciu inequalities fordeterminants, permanents, and generalized matrix functions. The new operatorinequalities and their corollaries contain a few recently publishedinequalities on positive definite matrices as special cases.

Hlawka-Popoviciu inequalities on positive definite tensors