Let \(X_1,\ldots , X_{d+2}\) be random points in \(\mathbb {R}^d\) . The classical Sylvester problem asks to determine the probability that the convex hull of these points, denoted by \(P:= [X_1,\ldots , X_{d+2}]\) , is a simplex. In the present paper, we study a refined version of this problem which asks to determine the probability that P has a given combinatorial type. It is known that there are \(\lfloor d/2\rfloor +1\) possible combinatorial types of simplicial d-dimensional polytopes with at most \(d+2\) vertices. These types are denoted by \(T_0^d, T_1^d, \ldots , T_{\lfloor d/2 \rfloor }^d\) , where \(T_0^d\) is a simplex with \(d+1\) vertices, while the remaining types have exactly \(d+2\) vertices. Our aim is thus to compute the probability \( p_{d,m} := \mathbb {P}[P \text { is of type } T_{m}^d], \qquad m\in \{0,1,\ldots , \lfloor d/2 \rfloor \}. \) The classical Sylvester problem corresponds to the case \(m=0\) . We shall compute \(p_{d,m}\) for all m in the following cases: (a) \(X_1,\ldots , X_{d+2}\) are i.i.d. normal; (b) \(X_1,\ldots , X_{d+2}\) follow a d-dimensional beta or beta prime distribution, which includes the uniform distribution on the ball or on the sphere as special cases; (c) \(X_1,\ldots , X_{d+2}\) form a random walk with exchangeable increments. As a by-product of case (a) we recover a recent solution to Youden’s demon problem which asks to determine the probability that, in a one-dimensional i.i.d. normal sample \(\xi _1,\ldots , \xi _n\) , the empirical mean \(\frac{1}{n} (\xi _1 + \ldots + \xi _n)\) lies between the k-th and the \((k+1)\) -st order statistics. We also consider the conic (or spherical) version of the refined Sylvester problem and solve it in several special cases.
