The largest small n-dimensional polytope with n+3 vertices

Volume of Convex Hull of Two Bodies and Related Problems

“…For polytopes with (n+2) vertices this question was answered by Kind and Kleinschmidt [29]. The solution for polytopes with n + 3 vertices was published in [30], which later turned out to be incomplete (cf. [31]), and thus, this case is still open.…”

“…The solution for polytopes with n + 3 vertices was published in [30], which later turned out to be incomplete (cf. [31]), and thus, this case is still open. We mention two problems in more detail:…”

Discrete isoperimetric problems in spaces of constant curvature

“…Much less is known about convex polytopes with a given number of vertices in the d-dimensional Euclidean space E d . Among the known results for polytopes with d + 2 vertices, which in most problems seems to be the first interesting case, we can mention the paper [5] of Böröczky and Böröczky Jr. finding the minimum surface area polytopes of unit volume in this family, [15] of Klein and Wessler determining the maximum volume polytopes with unit diameter, and [10] of G.Horváth and the second named author about maximum volume polytopes with unit circumradius. We note that both [15] and [10] contain partial results for polytopes with d + 3 vertices (see also [16]).…”

Maximum volume polytopes inscribed in the unit sphere

“…For polytopes with (d + 2) vertices this question was answered by Kind and Kleinschmidt [13]. The solution for polytopes with d + 3 vertices was published in [14], which later turned out to be incomplete (cf. [15]), and thus, this case is still open.…”

“…The solution for polytopes with d + 3 vertices was published in [14], which later turned out to be incomplete (cf. [15]), and thus, this case is still open.…”