  We study the geometry of the image of the nonnegative orthant under the
power-sum map and the elementary symmetric polynomials map. After analyzing the
image in finitely many variables, we concentrate on the limit as the number of
variables approaches infinity. We explain how the geometry of the limit plays a
crucial role in undecidability results in nonnegativity of symmetric
polynomials, deciding validity of trace inequalities in linear algebra, and
extremal combinatorics - recently observed by Blekherman, Raymond, and F. Wei.
We verify the experimental observation that the image has the combinatorial
geometry of a cyclic polytope made by Mel\'anov\'a, Sturmfels, and Winter, and
generalize results of Choi, Lam, and Reznick on nonnegative even symmetric
polynomials. We also show that undecidability does not hold for the normalized
power sum map.
