No abstract

We consider the space [0, n] 3 , imagined as a three dimensional, axis-aligned grid world partitioned into n 3 1 × 1 × 1 unit cubes. Each cube is either considered to be empty, in which case a line of sight can pass through it, or obstructing, in which case no line of sight can pass through it. From a given position, some of these obstructing cubes block one's view of other obstructing cubes, leading to the following extremal problem: What is the largest number of obstructing cubes that can be simultaneously visible from the surface of an observer cube, over all possible choices of which cubes of [0, n] 3 are obstructing? We construct an example of a configuration in which Ω n 8 3 obstructing cubes are visible, and generalize this to an example with Ω n d− 1 d visible obstructing hypercubes for dimension d > 3. Using Fourier analytic techniques, we prove an O n d− 1 d log n upper bound in a reduced visibility setting.

We discuss critical loci of convex symmetric domains in the plane and give a construction which, in particular, furnishes examples of domains having critical locus of fractional Hausdorff dimension.