On the minimal volume of simplices enclosing a convex body

On some determinant and matrix inequalities with a geometrical flavour

“…Geometrically, the right side of (1.8) relates to the maximal volume of n-simplex whose vertices are in E. The relationship between the maximal volume of the n-simplex whose vertices are in E and the measure of E has been studied before (see [10], [13]). It is well known that by compactness given a compact convex set E ⊂ R n , there exists a simplex T ⊂ E of maximal volume.…”

“…Since F is an arbitrary facet of T , T is contained in the simplex −n(T − c) + c, where c is the centroid of T . See [10] for details. So T ⊂ E ⊂ −n(T − c) + c, and thus |E| ≤ n n |T |.…”

“…So fix all columns except the 3rd column, we have s ≥ sup (x 10 ) 3 ∈E As before, by (3.5) we assume there exists T 0 coE x 10 1 is an ellipsoid with principal axes parallel to the coordinate axes in Since T 0 coE x 10 1 is an axis-parallel rectangle in R 3 , it can be written as A 1 × A 2 × A 3 , where A 1 , A 2 , A 3 are intervals in R. Similar to the proof of (2.10) together with S(T 0 coE x 10 1 ) = S(T 0 coE x 10…”

“…= (x 10 ) 1 (x 10 ) 2 (x 10 ) 3 ∈ E 1 with (x 10 ) 1 (x 10 ) 2 = x 10 ∈ M 3×2 and (x 10 ) 3 ∈ E x10 1 ; A 2 := (x 21 ) 1 (x 21 ) 2 (x 21 ) 3 ∈ E 2 with the condition (x 21 ) 1 = x 21 ∈ M 3×1 and (x 21 ) 2 ∈ F x21 2 .…”

“…R 3 . From(3.6) we may assume T 0 coE x10 1 is an axis-parallel rectangle. Because of the invariance under O(x 10 ) 1 + (x 21 ) 1 x 2 + (x 10 ) 2 + (x 21 ) 2 x 3 + (x 10 ) 3 + (x 21 ) 3 | det T 0 (x 1 + (x 10 ) 1 + (x 21 ) 1 ) T 0 (x 2 + (x 10 ) 2 + (x 21 ) 2 ) T 0 (x 3 + (x 10 ) 3 + (x 21 ) 3 ) |.…”