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Hand-intensive manufacturing processes, such as composite layup and textile draping, require significant human dexterity to accommodate task complexity. These strenuous hand motions often lead to musculoskeletal disorders and rehabilitation surgeries. We develop a data-driven ergonomic risk assessment system with a special focus on hand and finger activity to better identify and address ergonomic issues related to hand-intensive manufacturing processes. The system comprises a multi-modal sensor testbed to collect and synchronize operator upper body pose, hand pose and applied forces; a Biometric Assessment of Complete Hand (BACH) formulation to measure high-fidelity hand and finger risks; and industry-standard risk scores associated with upper body posture, RULA, and hand activity, HAL. Our findings demonstrate that BACH captures injurious activity with a higher granularity in comparison to the existing metrics. Machine learning models are also used to automate RULA and HAL scoring, and generalize well to unseen participants. Our assessment system, therefore, provides ergonomic interpretability of the manufacturing processes studied, and could be used to mitigate risks through minor workplace optimization and posture corrections., Comment: 26 pages, 7 figures

A basic version of the P\'olya-Szeg\H{o} inequality states that if $\Phi$ is a Young function, the $\Phi$-Dirichlet energy -- the integral of $\Phi(\|\nabla f\|)$ -- of a suitable function $f\in \mathcal{V}(\mathbb{R}^n)$, the class of nonnegative measurable functions on $\mathbb{R}^n$ that vanish at infinity, does not increase under symmetric decreasing rearrangement. This fact, along with variants that apply to polarizations and to Steiner and certain other rearrangements, has numerous applications. Very general versions of the inequality are proved that hold for all smoothing rearrangements, those that do not increase the modulus of continuity of functions. The results cover all the main classes of functions previously considered: Lipschitz functions $f\in \mathcal{V}(\mathbb{R}^n)$, functions $f\in W^{1,p}(\mathbb{R}^n)\cap\mathcal{V}(\mathbb{R}^n)$ (when $1\le p<\infty$ and $\Phi(t)=t^p$), and functions $f\in W^{1,1}_{loc}(\mathbb{R}^n)\cap\mathcal{V}(\mathbb{R}^n)$. In addition, anisotropic versions of these results, in which the role of the unit ball is played by a convex body containing the origin in its interior, are established. Taken together, the results bring together all the basic versions of the P\'olya-Szeg\H{o} inequality previously available under a common and very general framework., Comment: 49 pages

Two related questions are discussed. The first is when reflection symmetry in a finite set of $i$-dimensional subspaces, $i\in \{1,\dots,n-1\}$, implies full rotational symmetry, i.e., the closure of the group generated by the reflections equals $O(n)$. For $i=n-1$, this has essentially been solved by Burchard, Chambers, and Dranovski, but new results are obtained for $i\in \{1,\dots,n-2\}$. The second question, to which an essentially complete answer is given, is when (full) rotational symmetry with respect to a finite set of $i$-dimensional subspaces, $i\in \{1,\dots,n-2\}$, implies full rotational symmetry, i.e., the closure of the group generated by all the rotations about each of the subspaces equals $SO(n)$. The latter result also shows that a closed set in $\mathbb{R}^n$ that is invariant under rotations about more than one axis must be a union of spheres with their centers at the origin., Comment: 16 pages, 2 figures. The core of this paper was Section 3 of arXiv:1908.03259v2 and will be published separately following the suggestion of a referee. Some proofs are shorter and clearer (Lemma 3.1 and Theorem 3.2), others are completely different and much shorter (the symmetry extension lemma, now Lemma 3.7). In version 2 some typos have been corrected

The paper has two main goals. The first is to take a new approach to rearrangements on certain classes of measurable real-valued functions on $\mathbb{R}^n$. Rearrangements are maps that are monotonic (up to sets of measure zero) and equimeasurable, i.e., they preserve the measure of super-level sets of functions. All the principal known symmetrization processes for functions, such as Steiner and Schwarz symmetrization, are rearrangements, and these have a multitude of applications in diverse areas of the mathematical sciences. The second goal is to understand which properties of rearrangements characterize polarization, a special rearrangement that has proved particularly useful in a number of contexts. In order to achieve this, new results are obtained on the structure of measure-preserving maps on convex bodies and of rearrangements generally., Comment: 44 pages. In version 2 two minor mistakes in the proofs of Lemma 6.3 and Theorem 6.6 have been corrected. Version 3 is the one accepted for publication in Advances in Mathematics

Steiner and Schwarz symmetrizations, and their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational symmetrizations, are investigated. The focus is on the convergence of successive symmetrals with respect to a sequence of $i$-dimensional subspaces of $\mathbb{R}^n$. Such a sequence is called universal for a family of sets if the successive symmetrals of any set in the family converge to a ball with center at the origin. New universal sequences for the main symmetrizations, for all valid dimensions $i$ of the subspaces, are found, by combining two groups of results. The first, published separately, provides finite sets ${\mathcal{F}}$ of subspaces such that reflection symmetry (or rotational symmetry) with respect to each subspace in ${\mathcal{F}}$ implies full rotational symmetry. In the second, proved here, a theorem of Klain for Steiner symmetrization is extended to Schwarz, Minkowski, Minkowski-Blaschke, and fiber symmetrizations, showing that if a sequence of subspaces is drawn from a finite set ${\mathcal{F}}$ of subspaces, the successive symmetrals of any compact convex set converge to a compact convex set that is symmetric with respect to any subspace in ${\mathcal{F}}$ appearing infinitely often in the sequence. It is also proved that for Steiner, Schwarz, and Minkowski symmetrizations, a sequence of $i$-dimensional subspaces is universal for the class of compact sets if and only if it is universal for the class of compact convex sets, and Klain's theorem is shown to hold for Schwarz symmetrization of compact sets., Comment: 29 pages. changes: following the suggestion of a referee, Section 3 of version 2 has been removed and has become the core of a new paper by the same authors titled "Full rotational symmetry from reflections or rotational symmetries in finitely many subspaces", also submitted to arXiv

The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when $G$ and $\psi$ are both increasing, thus complementing our previous work. The Minkowski problem asks to characterize Borel measures $\mu$ on $\sphere$ for which there is a convex body $K$ in $\R^n$ containing the origin such that $\mu$ equals $\deV(K, \cdot)$, up to a constant. A major step in the analysis concerns discrete measures $\mu$, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. For general (not necessarily discrete) measures $\mu$, we use an approximation argument. This approach is also applied to the case where $G$ is decreasing and $\psi$ is increasing, and hence augments our previous work. When the measures $\mu$ are even, solutions that are origin-symmetric convex bodies are also provided under some mild conditions on $G$ and $\psi$. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when $\mu$ is discrete or even.