Über das Löwnersche Ellipsoid und sein Analogon unter den einem Eikörper einbeschriebenen Ellipsoiden

On some mean values associated with a randomly selected simplex in a convex set

“…Let us denote by J%Γ the class of all convex bodies for which these two conditions are satisfied. If a number μ is defined by μ = inΐm r (K) then μ has obviously the property that for every (4) μâ nd that there exists a sequence K u K 2 , of convex bodies in 3ίί such that ( 5) lim m r (Ki) = μ .…”

“…It is known (see Danzer, Laugwitz, and Lenz [5]) that there is an ellipsoid, say L, which contains K and has smallest possible volume. It is also known (see Hadwiger [7], p 170) that there is a sequence of planes, say H l9 H 2 , * ,in R n such that the sequence of convex bodies which is defined by K t -K, K i+ι -K^Hi) (i -1, 2, •) contains a subsequence that converges to a sphere S. It follows that there are volume preserving affine transformations σ u σ 2 , such that the sequence σ λ K, σ 2 K, converges to S. If K = L the proof of the lemma is obviously finished.…”

Symmetry of convex sets and its applications to the extremal ellipsoids of convex bodies

“…The uniqueness of CE(K) also follows from the famous paper of John [18] although he does not state it explicitly. Subsequently, Danzer, Laugwitz, and Lenz [11], and Zaguskin [39] prove the uniqueness of both ellipsoids. The inscribed ellipsoid problem IE(K) is often called John ellipsoid, and sometimes Löwner-John ellipsoid, especially in Banach space geometry literature.…”

“…They also have applications in differential geometry [23], Lie group theory [11], and symplectic geometry, among others. The ellipsoid algorithm of Khachiyan [19] for linear programming sparked general interest of optimizers in the circumscribed ellipsoid problem.…”

Some applications of means of convex bodies

“…We allow x to vary and so generate two collections of ellipsoids {E (A v (x))} and {EiB^x))}. For v = 0 Danzer, Laugwitz and Lenz in [4] have shown that in the first collection there is a unique ellipsoid for which the volume W Q is a minimum and in the second collection there is a unique ellipsoid for which the volume is a maximum. We have not been able to decide if this is also true for v = l r 2,* , n -1 with W v in place of the volume.…”