In the seventh dimension, you have access to the possible worlds that start with different initial conditions. In theory, if you could master the fifth and sixth dimension, you could travel back in time or go to different futures. In the sixth, we would see a plane of possible worlds, where we could compare and position all the possible universes that start with the same initial conditions as this one (i.e. The other dimensions are where the deeper possibilities come into play, and explaining their interaction with the others is where things get particularly tricky for physicists. Along with the three other dimensions, knowing an objects position in time is essential to plotting its position in the universe. Scientists believe that the fourth dimension is time, which governs the properties of all known matter at any given point. Beyond these three lie the seven dimensions which are not immediately apparent to us, but which can be still be perceived as having a direct effect on the universe and reality as we know it. The perfect example of this is a cube, which exists in three dimensions and has a length, width, depth, and hence volume. The third dimension involves depth (the z-axis), and gives all objects a sense of area and a cross-section. Add to it a second dimension, the y-axis (or height), and you get an object that becomes a 2-dimensional shape (like a square). A good description of a one-dimensional object is a straight line, which exists only in terms of length and has no other discernible qualities. The first dimension, as already noted, is that which gives it length (aka. These different aspects are what govern the universe, the fundamental forces of nature, and all the elementary particles contained within. In fact, the theoretical framework of Superstring Theory posits that the universe exists in ten different dimensions. We are immediately aware of the three dimensions that surround us on a daily basis – those that define the length, width, and depth of all objects in our universes (the x, y, and z axes, respectively).īeyond these three visible dimensions, scientists believe that there may be many more. This generalizes the concept of “reach”: the reach of a set A can be deﬁned sup are Cebysev in Od.To break it down, dimensions are simply the different facets of what we perceive to be reality. As a generalization of the ˇ Cebyˇ sev property, we deﬁne A to be ˇ Cebyˇ sev relative to X 0 if every point in X 0 has a unique nearest point in A. In particular, in Euclidean spaces, the ˇ Cebyˇ sev sets are precisely those that are nonempty, closed and convex. For Minkowski spaces (ﬁnite dimensional Banach spaces), every ˇ Cebyˇ sev set is convex if balls are smooth, while if the balls are strictly convex, every nonempty closed convex set is ˇ Cebyˇ sev. For such spaces, the ˇ Cebyˇ sev property is related to convexity. This property has been studied extensively for normed linear spaces. For any metric space ( X, ρ ), we deﬁne a set A ⊂ X to be a ˇ Cebyˇ sev set (“be ˇ Cebyˇ sev” or “have the ˇ Cebyˇ sev property”) if for every x ∈ X there is a unique nearest point in A. They are constructed as translational closures of appropriate nested arcs.

Some new families of ˇ Cebyˇ sev sets in hyperspaces are exhibited, with dimension d + 1 (where d is the dimension of the underlying space). In particular, in hyperspaces over normed linear spaces several quite diﬀerent classes of ˇ Cebyˇ sev sets are known, with no unifying description. In Euclidean spaces, this property is equivalent to being closed, convex, and nonempty, but in other spaces classiﬁcation of ˇ Cebyˇ sev sets may be sig- niﬁcantly more diﬃcult. A ˇ Cebyˇ sev set in a metric space is one such that every point of the space has a unique nearest neighbour in the set. DAWSON Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia, B3H 3C3, Canada E-mail: Abstract. **************************************** BANACH CENTER PUBLICATIONS, VOLUME ** INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 200* SOME ˇ CEBY ˇ SEV SETS WITH DIMENSION D +1 IN HYPERSPACES OVER R D ROBERT J.

Dimension of convex hyperspaces