Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes

Abstract

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope that is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants. Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak’s dual Brunn–Minkowski theory.

The Maximum Surface Area Polyhedron with Five Vertices Inscribed in the Sphere $\mathbb{S}^2$

“…From results in [22,21,24,34,35] it follows that as the dimension tends to infinity, the ratio of the best inscribed to best circumscribed approximation of the ball is of the order O(d −1 ln(d)). (The same result holds true for approximation with respect to any intrinsic volume [6]). ) This is some evidence the answer is affirmative.…”

“…When v = 6, there are 4 possibilities for (v, e, f ); Steinitz [42] showed that a polytope in R 3 has v vertices, e edges and f facets if and only if the vector (v, e, f ) satisfies the Euler relation v−e+f = 2 and the inequalities v ≤ 2f − 4 and f ≤ 2v − 4. Hence, if v = 6 then 5 ≤ f ≤ 8 and e = f + 4, so the possible face vectors (v, e, f ) are (6,9,5), (6,10,6), (6,11,7) and (6,12,8). Checking cases, we find that G(6, 12, 8) has the maximum value among the four possibilities.…”

Perimeter Approximation of Convex Discs in the Hyperbolic Plane and on the Sphere

“…We note that approximations of convex bodies in d-dimensional Euclidean space with respect to all intrinsic volume deviations, where the relative position of the polytope and body is not restricted, have recently been studied by Besau et al [2]. They prove asymptotic estimates for best approximations of the unit ball in these deviations measures.…”

“…)) as δ → 0 + . Thus, by (2), the arc of bd K between o and the positive intersection point of l and bd K has length…”

Random inscribed polytopes in projective geometries

“…The dual Brunn-Minkowski theory, introduced by Lutwak [40,42], is a variant of classical Brunn-Minkowski theory, which has become a central piece of modern convex geometry, see e.g. [2,7,14,25,26,31,43]. Its starting point is the replacement of Minkowski sum by the so-called radial sum of convex bodies, or more generally, star bodies.…”