During the Spring 2023 ICERM semester, we initiated a parametric study of hyperplane sections of polytopes. Our framework enables the resolution of various optimization problems, which we will discuss in detail. However, when extending the study to nonlinear convex bodies, the combinatorial approach proves inadequate. We explore two alternative methods to address analogous optimization problems for slices of more general bodies, such as semialgebraic convex bodies and polygauge convex bodies. This talk is based on joint works with Marie-Charlotte Brandenburg, Jesús A. De Loera, Jared Miller, Matteo Tacchi, and Mauricio Velasco.