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23 –24], E i # E 0 for i ¼ 1, . . , p if and only if there exists non-negative scalars t1 , . . , tp such that T0 (x) À ti Ti (x) 0, i ¼ 1, . . , p, or equivalently, such that 2 F0 4 gT0 0 g0 À1 g0 3 2 0 Fi gT0 5 À ti 4 gTi ÀF0 0 gi hi 0 3 0 05 0 0, for i ¼ 1, . . , p: We can ﬁnd the MVE containing the union of ellipsoids E 1 , . . , E p by solving the matrix completion problem: minimize log det F0À1 F0 . 0, t1 ! 0, . . , tp ! 0, 3 2 g0 0 Fi gi 6 T T7 À1 g 0 5 À t i 4 gi hi subject to 2 F0 6 T 4 g0 0 g0 ÀF0 0 0 3 0 7 05 0, 0 for i ¼ 1, .