On quasiconvexity of precompact-subset spaces

Suppose $X$ is a metric space and $BCl(X)$ the collection of its bounded closed subsets as a metric space with respect to Hausdorff distance (and call $BCl(X)$ the \emph{bounded-subset space} of $X$). The question of whether or not one can characterize (the existence of) a rectifiable path in some subspace $\mathcal{J}$ of $BCl(X)$ entirely in terms of rectifiable paths in $X$ does not seem to have been given serious consideration. In this paper, we make some progress with the case where $\mathcal{J}$ consists of precompact subsets of $X$ (with such a $\mathcal{J}$ called a \emph{precompact-subset space} of $X$). Specifically, in certain precompact-subset spaces $\mathcal{J}$ of $X$, we give a criterion to determine (the existence of) a rectifiable path in $\mathcal{J}$ using rectifiable paths in $X$. We then show that certain path connectivity properties, especially quasiconvexity, inherited from $X$ by such precompact-subset spaces of $X$ can be determined in an automatic way using our criterion. Meanwhile, we also give a concise review of our earlier work on quasiconvexity of \emph{finite-subset spaces} of $X$

Metric Geometry of Finite Subset Spaces

If X is a (topological) space, the nth finite subset space of X, denoted by X(n), consists of n-point subsets of X (i.e., nonempty subsets of cardinality at most n) with the quotient topology induced by the unordering map q:X^n--\u3e X(n), (x_1,...,x_n)--\u3e{x_1,...,x_n}. That is, a subset A of X(n) is open if and only if its preimage under q is open in the product space X^n. Given a space X, let H(X) denote all homeomorphisms of X. For any subclass C of homeomorphisms in H(X), the C-geometry of X refers to the description of X up to homeomorphisms in C. Therefore, the topology of X is the H(X)-geometry of X. By a (C-) geometric property of X we will mean a property of X that is preserved by homeomorphisms of X (in C). Metric geometry of a space X refers to the study of geometry of X in terms of notions of metrics (e.g., distance, or length of a path, between points) on X. In such a study, we call a space X metrizable if X is homeomorphic to a metric space. Naturally, X(n) always inherits some aspect of every geometric property of X or X^n. Thus, the geometry of X(n) is in general richer than that of X or X^n. For example, it is known that if X is an orientable manifold, then (unlike X^n) X(n) for n\u3e1 can be an orientable manifold, a non-orientable manifold, or a non-manifold. In studying geometry of X(n), a central research question is ``If X has geometric property P, does it follow that X(n) also has property P?\u27\u27. A related question is If X and Y have a geometric relation R, does it follow that X(n) and Y(n) also have the relation R? . Extensive work exists in the literature on the richness of the geometry of X(n). Nevertheless, despite the fact that the spaces X(n) considered in those investigations are metrizable (which is the case if and only if X is itself metrizable) the important role of metrics has been mostly ignored. Consequently, the existing results mostly elucidate topological aspects of the geometry of X(n). The main goal of this thesis is to attempt to answer the above research question(s) for several geometric properties, with metrics playing a significant role (hence the title phrase Metric Geometry of ... ). Some of the questions are relatively easy and will be answered completely. However, a question such as If a normed space X is an absolute Lipschitz retract, does it follow that X(n) is also an absolute Lipschitz retract? appears to require considerable effort and will be answered only partially. By the definition of an absolute Lipschitz retract, establishing the existence of Lipschitz retractions X(n) --\u3e X(n-1) for all n\u3e= 1 would be a partial positive answer to this question. Among other things, we will prove the following. If X is a metrizable space, then so is X(n). If a metric space X is a snowflake, quasiconvex, or doubling then so is X(n). If two spaces X and Y are (Lipschitz) homotopy equivalent, then so are X(n) and Y(n). If X is a normed space (which is Lipschitz k-connected for all k\u3e= 0), then X(n) is Lipschitz k-connected for all k\u3e= 0. If X is a normed space, there exist (i) Holder retractions X(n)--\u3e X(n-1), (ii) Lipschitz retractions X(n)--\u3e X(1),X(2), and (iii) Lipschitz retractions X(n)--\u3e X(n-1) when the dimension of X is finite or X is a Hilbert space

Optimal Inference for Distributed Detection

In distributed detection, there does not exist an automatic way of generating optimal decision strategies for non-affine decision functions. Consequently, in a detection problem based on a non-affine decision function, establishing optimality of a given decision strategy, such as a generalized likelihood ratio test, is often difficult or even impossible. In this thesis we develop a novel detection network optimization technique that can be used to determine necessary and sufficient conditions for optimality in distributed detection for which the underlying objective function is monotonic and convex in probabilistic decision strategies. Our developed approach leverages on basic concepts of optimization and statistical inference which are provided in appendices in sufficient detail. These basic concepts are combined to form the basis of an optimal inference technique for signal detection. We prove a central theorem that characterizes optimality in a variety of distributed detection architectures. We discuss three applications of this result in distributed signal detection. These applications include interactive distributed detection, optimal tandem fusion architecture, and distributed detection by acyclic graph networks. In the conclusion we indicate several future research directions, which include possible generalizations of our optimization method and new research problems arising from each of the three applications considered