Minimal simplices inscribed in a convex body

Abstract

Measuring how far a convex body \(\mathcal{K }\) (of dimension \(n\)) with a base point \({O}\in \,\text{ int }\,\mathcal{K }\) is from an inscribed simplex \(\Delta \ni {O}\) in “minimal” position, the interior point \({O}\) can display regular or singular behavior. If \({O}\) is a regular point then the \(n+1\) chords emanating from the vertices of \(\Delta \) and meeting at \({O}\) are affine diameters, chords ending in pairs of parallel hyperplanes supporting \(\mathcal{K }\). At a singular point \({O}\) the minimal simplex \(\Delta \) degenerates. In general, singular points tend to cluster near the boundary of \(\mathcal{K }\). As connection to a number of difficult and unsolved problems about affine diameters shows, regular points are elusive, often non-existent. The first result of this paper uses Klee’s fundamental inequality for the critical ratio and the dimension of the critical set to obtain a general existence for regular points in a convex body with large distortion (Theorem A). This, in various specific settings, gives information about the structure of the set of regular and singular points (Theorem B). At the other extreme when regular points are in abundance, a detailed study of examples leads to the conjecture that the simplices are the only convex bodies with no singular points. The second and main result of this paper is to prove this conjecture in two different settings, when (1) \(\mathcal{K }\) has a flat point on its boundary, or (2) \(\mathcal{K }\) has \(n\) isolated extremal points (Theorem C).