A polytrope is a tropical polyhedron that is also classically convex. We study the tropical combinatorial types of polytropes associated to weighted directed acyclic graphs (DAGs). This family of polytropes arises in algebraic statistics when describing the model class of max-linear Bayesian networks. We show how the edge weights of a network directly relate to the facet structure of the corresponding polytrope. We also give a classification of polytropes from weighted DAGs at different levels of equivalence. These results give insight on the statistical problem of identifiability for a max-linear Bayesian network.

The Fréchet mean is a fundamental notion of central tendency defined as a minimizer of a sum of squared distances in a general metric space. In this paper, we study Fréchet means in tropical geometry—a piecewise linear, combinatorial, and polyhedral variant of algebraic geometry—by formulating and solving the associated tropical quadratic optimization problem. We give a geometric characterization of the collection of all tropical Fréchet means as a bounded set that is simultaneously tropically and classically convex, hence a polytrope. We establish the existence of positivity certificates for maxima of finitely many quadratic polynomials in R [ x 1 , … , x n ] whose homogeneous quadratic components are sums of squares, which provides a symbolic framework for exact optimization. Using this structure, we develop algorithms for computing tropical Fréchet means and the associated Fréchet mean polytrope. We further describe a combinatorial type decomposition of the objective function induced by braid arrangements, yielding a piecewise quadratic representation and a fully symbolic method for exact computation.

The Fréchet mean is a key measure of central tendency as a barycenter for a given set of points in a general metric space. It is computed by solving an optimization problem and is a fundamental quantity in statistics. In this paper, we study Fréchet means in tropical geometry—a piecewise linear, combinatorial, and polyhedral variant of algebraic geometry that has gained prominence in applications. A key property of Fréchet means is that uniqueness is generally not guaranteed, which is true in tropical settings. In solving the tropical Fréchet mean optimization problem, we obtain a geometric characterization of the collection of all Fréchet means in a general tropical space as a tropically and classically convex polytope. Furthermore, we prove that a certificate of positivity for finitely many quadratic polynomials in \(\mathbb {R}[x_1,\ldots ,x_n]\) always exists, given that their quadratic homogeneous components are sums of squares. We propose an algorithm to symbolically compute the Fréchet mean polytope based on our exact quadratic optimization result and study its complexity.

The vibrotactile funneling illusion is the sensation of a single (non-existing) stimulus somewhere in-between the actual stimulus locations. Its occurrence depends upon body location, distance between the actuators, signal synchronization, and intensity. Related work has shown that the funneling illusion may occur on the forehead. We were able to reproduce these findings and explored five further regions to get a more complete picture of the occurrence of the funneling illusion on the head. The results of our study (24 participants) show that the actuator distance, for which the funneling illusion occurs, strongly depends upon the head region. Moreover, we evaluated the centralizing bias (smaller perceived than actual actuator distances) for different head regions, which also showed widely varying characteristics. We computed a detailed heat map of vibrotactile localization accuracies on the head. The results inform the design of future tactile head-mounted displays that aim to support the funneling illusion.