v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
292 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.3.0.0.1 Example. Triangle.Consider the EDM in (694), but missing one of its entries:⎡0⎤1 d 13D = ⎣ 1 0 4 ⎦ (695)d 31 4 0Can we determine unknown entries of D by applying the metric properties?Property 1 demands √ d 13 , √ d 31 ≥ 0, property 2 requires the main diagonalbe 0, while property 3 makes √ d 31 = √ d 13 . The fourth property tells us1 ≤ √ d 13 ≤ 3 (696)Indeed, described over that closed interval [1, 3] is a family of triangularpolyhedra whose angle at vertex x 2 varies from 0 to π radians. So, yes wecan determine the unknown entries of D , but they are not unique; nor shouldthey be from the information given for this example.5.3.0.0.2 Example. Small completion problem, I.Now consider the polyhedron in Figure 74(b) formed from an unknownlist {x 1 ,x 2 ,x 3 ,x 4 }. The corresponding EDM less one critical piece ofinformation, d 14 , is given by⎡⎤0 1 5 d 14D = ⎢ 1 0 4 1⎥⎣ 5 4 0 1 ⎦ (697)d 14 1 1 0From metric property 4 we may write a few inequalities for the two trianglescommon to d 14 ; we find√5−1 ≤√d14 ≤ 2 (698)We cannot further narrow those bounds on √ d 14 using only the four metricproperties (5.8.3.1.1). Yet there is only one possible choice for √ d 14 becausepoints x 2 ,x 3 ,x 4 must be collinear. All other values of √ d 14 in the interval[ √ 5−1, 2] specify impossible distances in any dimension; id est, in thisparticular example the triangle inequality does not yield an interval for √ d 14over which a family of convex polyhedra can be reconstructed.
5.3. ∃ FIFTH EUCLIDEAN METRIC PROPERTY 293x 3x 4x 3x 4(a)√512(b)1x 1 1 x 2x 1 x 2Figure 74: (a) Complete dimensionless EDM graph. (b) Emphasizingobscured segments x 2 x 4 , x 4 x 3 , and x 2 x 3 , now only five (2N −3) absolutedistances are specified. EDM so represented is incomplete, missing d 14 asin (697), yet the isometric reconstruction (5.4.2.2.5) is unique as proved in5.9.3.0.1 and5.14.4.1.1. First four properties of Euclidean metric are nota recipe for reconstruction of this polyhedron.We will return to this simple Example 5.3.0.0.2 to illustrate more elegantmethods of solution in5.8.3.1.1,5.9.3.0.1, and5.14.4.1.1. Until then, wecan deduce some general principles from the foregoing examples:Unknown d ij of an EDM are not necessarily uniquely determinable.The triangle inequality does not produce necessarily tight bounds. 5.4Four Euclidean metric properties are insufficient for reconstruction.5.4 The term tight with reference to an inequality means equality is achievable.
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292 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.3.0.0.1 Example. Triangle.Consider the EDM in (694), but missing one of its entries:⎡0⎤1 d 13D = ⎣ 1 0 4 ⎦ (695)d 31 4 0Can we determine unknown entries of D by applying the metric properties?Property 1 demands √ d 13 , √ d 31 ≥ 0, property 2 requires the main diagonalbe 0, while property 3 makes √ d 31 = √ d 13 . The fourth property tells us1 ≤ √ d 13 ≤ 3 (696)Indeed, described over that closed interval [1, 3] is a family of triangularpolyhedra whose angle at vertex x 2 varies from 0 to π radians. So, yes wecan determine the unknown entries of D , but they are not unique; nor shouldthey be from the information given for this example.5.3.0.0.2 Example. Small completion problem, I.Now consider the polyhedron in Figure 74(b) formed from an unknownlist {x 1 ,x 2 ,x 3 ,x 4 }. The corresponding EDM less one critical piece ofinformation, d 14 , is given by⎡⎤0 1 5 d 14D = ⎢ 1 0 4 1⎥⎣ 5 4 0 1 ⎦ (697)d 14 1 1 0From metric property 4 we may write a few inequalities for the two trianglescommon to d 14 ; we find√5−1 ≤√d14 ≤ 2 (698)We cannot further narrow those bounds on √ d 14 using only the four metricproperties (5.8.3.1.1). Yet there is only one possible choice for √ d 14 becausepoints x 2 ,x 3 ,x 4 must be collinear. All other values of √ d 14 in the interval[ √ 5−1, 2] specify impossible distances in any dimension; id est, in thisparticular example the triangle inequality does not yield an interval for √ d 14over which a family of convex polyhedra can be reconstructed.