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A drawback of these centers is that they generally cannot be computed cost-e ectively. For the ellipsoid method, its advantage in not keeping knowledge of the cutting planes is also a disadvantage to practical e ciency for solving certain 46 CHAPTER 2. GEOMETRY OF CONVEX INEQUALITIES problems, such as linear programs. Thus, another type of center, called the analytic center for a convex polyhedron given by linear inequalities, was introduced. , = fy 2 Rm : c ? AT y 0g; where A 2 Rm n and c 2 Rn are given and A has rank m.

Various centers were considered as test points. In this chapter, we review these centers and their associated measures. We show that, similar to these center-section algorithms, interior-point algorithms use a new measure of the containing set represented by linear inequalities. This measure is \analytic", and is relatively easy to compute. Its associated center is called the analytic center. 1 Convex Bodies and Ellipsoids A natural choice of the measure would be the volume of the convex body. Interest in measure of convex bodies dates back as far as the ancient Greeks and Chinese who computed centers, areas, perimeters and curvatures of circles, triangle, and polygons.

More speci cally, we have the following problem: If one inequality in , say the rst one, of c ? AT y 0 needs to be translated, change c1 ? aT1 y 0 to aT1 ya ? 6). Let + := fy : aT y a ? aT y 0; c ? aT y 0; j = 2; :::ng j 1 1 j a + and let y be the analytic center of . Then, the max-potential for the new convex polytope + is exp(B( + )) = (aT y a 1 Regarding B( ) and B( + ), n Y ? aT1 ya ) (cj ? aTj ya): j =2 we prove the following theorem. 2. 7 Let and + be de ned as the above. Then B( + ) B( ) ?