By Dmitri Burago, Yuri Burago, Sergei Ivanov

"Metric geometry" is an method of geometry in response to the inspiration of size on a topological area. This strategy skilled a truly quick improvement within the previous few a long time and penetrated into many different mathematical disciplines, comparable to workforce idea, dynamical structures, and partial differential equations. the target of this graduate textbook is twofold: to provide an in depth exposition of simple notions and methods utilized in the speculation of size areas, and, extra more often than not, to provide an ordinary advent right into a vast number of geometrical subject matters on the topic of the thought of distance, together with Riemannian and Carnot-Caratheodory metrics, the hyperbolic aircraft, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic areas, convergence of metric areas, and Alexandrov areas (non-positively and non-negatively curved spaces). The authors are likely to paintings with "easy-to-touch" mathematical items utilizing "easy-to-visualize" tools. The authors set a not easy target of creating the center elements of the publication available to first-year graduate scholars. such a lot new suggestions and techniques are brought and illustrated utilizing least difficult circumstances and warding off technicalities. The publication comprises many routines, which shape an essential component of exposition.

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**Sample text**

Our target polygon Q consists of n5 partition rectangles of width p and height 1 spaced dt apart, connected by a bar of the same dimensions as the source polygon’s bar. The ﬁrst partition rectangle’s left edge and last partition rectangle’s right edge are ﬂush with the ends of the bar. The illustration of both polygons are given in Fig. 1. Both polygons’ bars have the same area and the total area of the element rectangles equals the total area of the partition rectangles, so the polygons have the same area.

Set F = ∅. For each conﬁguration c = (I1 , I2 ; z) ∈ F , do Steps (a)–(c) that apply, and put the generated conﬁgurations in F . (a) [J1i (ρ) = ∅ ∧ J2i (ρ) = ∅] The problem instance is not ρ-feasible (no point can pierce Di (ρ)). Stop (b) [J1i (ρ) = ∅] If I1 ∩ J1i (ρ) = ∅ then convert c into (I1 ∩ J1i (ρ), I2 ; z) else convert it into (J1i (ρ), I2 ; z + 1) and add r(I1 ) into ℘(c). (c) [J2i (ρ) = ∅] If I2 ∩ J2i (ρ) = ∅ then convert c into (I1 , I2 ∩ J2i (ρ); z) else convert it into (I1 , J2i (ρ); z + 1) and add r(I2 ) into ℘(c).

3 Main Results Now that we have deﬁned the problems, we state our main results. Theorem 1. k-Piece Dissection is NP-hard. We do not know whether k-Piece Dissection is in NP (and thus is NPcomplete). We discuss the diﬃculty of showing containment in NP in Sect. 7. We also prove that the optimization version, Min Piece Dissection, is hard to approximate to within some constant ratio: Theorem 2. 2 Both results are based on essentially the same reduction, from 5-Partition for Theorem 1 or from Max 5-Partition for Theorem 2.