INTRODUCTION. Fifty years ago, R. Bellman asked a remarkable minimization question (see Bellman [7], [6, p. 133], [8]) that can be phrased as follows: A hiker is lost in a forest whose shape and dimensions are precisely known to him. What is the best path for him to follow to escape from the forest? Call a path an escape path if it eventually leads out of the forest no matter what the initial starting point or the relative orientations of the path and forest. To solve the "lost in a forest"

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... lem we must find the "best" escape path. Bellman proposed two different interpretations of "best," one in which the maximum time to escape is minimized, and one in which the expected time to escape is minimized. A third interpretation (see Croft, Falconer, and Guy [12, p. 40]) involves maximizing the probability of escape within a specified time period. Bellman asked about two situations in particular: (1) the case in which the region is the infinite strip between two parallel lines a known distance apart, and (2) the case in which the region is a half-plane and the hiker's distance from the boundary is known. If "best" is taken to mean that the maximum time to escape is minimized, both of Bellman's particular situations have been investigated: for the first the minimax (i.e., shortest) escape path was found by Zalgaller [45] in 1961, and for the second the minimax escape path was described by Isbell [21] in 1957. In each of these two specific situations the shortest escape path is unique up to congruence. Little is known in either situation for other interpretations of "best." Our objective in this survey is to focus narrowly on the case in which the "best" escape path is the shortest one. We establish a fundamental connection between the "lost in a forest" problem and L. Moser's well-known "worm problem" (see Moser [31], Wetzel [41]), and we utilize a partial result in the worm problem due to Poole and Gerriets [35] to define a large class of regions for which the best escape path is a line segment. We give two examples for which the shortest escape path is not unique. And finally we summarize the little that is known for a variety of regions having various elementary geometric shapes. These problems can be phrased as "search" problems, in which one seeks the shortest search path to find the boundary of the region. For example, S. Burr (see Ogilvy [33, pp. 23-24, 149]) asked: A swimmer is lost in a dense fog at sea, and she knows the shape of the shore and her distance from it. What is the best path for her to follow to search for the shore? Williams [43] has included the "lost in a forest" problems in his recent list "Million Buck Problems" of unsolved problems of high potential impact on mathematics. A FEW GENERAL RESULTS. We begin by setting some language. A path y is a continuous and rectifiable mapping of [0, 1] into R2. The path y is oriented by increasing argument, from its initial point y (0) to its final point y (1). We write ?(y) for the length of the path y and {y} for its trace, i.e., its range {y(t) : 0 < t < 1}. We assume that a forest is a closed, convex region in the plane with nonempty interior. A path y is an escape path for a forest F if it meets the boundary 8F of F no October 2004] LOST IN A FOREST Clay Mathematics Institute, One Bow St., Cambridge, MA 02138 Steven. Finch @ inria.fr JOHN WETZEL did his undergraduate work at Purdue University and received a Ph.D. in mathematics from Stanford University in 1964, a student of Halsey Royden. He retired in 1999 from the University of Illinois at Urbana-Champaign after thirty-eight years of service. Always interested in classical geometry, he has most recently been studying the ways in which one shape fits in another-questions he regards as "fitting problems for retirement."