Geodesic
Approximation of Convex Compact Sets by Ellipsoids. Ellipsoids of Best Approximation
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control, Tome 262 (2008), pp. 103-126

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The problem of best approximation of a convex compact set in a finite-dimensional space by ellipsoids with respect to a special measure of deviation of an ellipsoid from a compact set is considered. An analytic description of ellipsoids of best approximation is given. 
@article{TM_2008_262_a8,
     author = {Yu. N. Kiselev},
     title = {Approximation of {Convex} {Compact} {Sets} by {Ellipsoids.} {Ellipsoids} of {Best} {Approximation}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {103--126},
     publisher = {mathdoc},
     volume = {262},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2008_262_a8/}
}
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Yu. N. Kiselev. Approximation of Convex Compact Sets by Ellipsoids. Ellipsoids of Best Approximation. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control, Tome 262 (2008), pp. 103-126. http://geodesic.mathdoc.fr/item/TM_2008_262_a8/