MathDB

Let

K

be a compact convex body in the

n

-dimensional Euclidean space. Let P_1,P_2,...,P_{n\plus{}1} be the vertices of a simplex having maximal volume among all simplices inscribed in

K

. Define the points P_{n\plus{}2},P_{n\plus{}3},... successively so that P_k \;(k>n\plus{}1) is a point of

K

for which the volume of the convex hull of

P_1,...,P_k

is maximal. Denote this volume by

V_k

. Decide, for different values of

n

, about the truth of the statement "the sequence V_{n\plus{}1},V_{n\plus{}2},... is concave." L. Fejes- Toth, E. Makai

Problem Statement