Problem 1
Let \(\mathrm{M}\) be the set of all points \(\mathrm{x}=\left(x_{1}, \mathrm{x}, \ldots, x_{n}, \ldots\right)\) in \(l_{2}\) satisfying the condition $$ \sum_{n=1}^{\infty} n^{2} x_{n}^{2} \leqslant 1 $$ Prove that \(\mathrm{M}\) is a convex set, but not a convex body.
Problem 3
We say that \(n+1\) points \(x_{1}, x_{2}, \ldots, x_{n+1}\) in a linear space \(\mathrm{L}\) are "in general position" if they do not belong to any \((\mathrm{n}-1)\)-dimensional subspace of \(\mathrm{L}\). The convex hull of a set of \(n+1\) points \(x_{1}, \mathrm{x}, \ldots, x_{n+1}\) in general position is called an \(\mathrm{n}\)-dimensional simplex, and the points \(x_{1}, x_{2}, \ldots\), \(x_{n+1}\) themselves are called the vertices of the simplex. Describe the zerodimensional, one-dimensional, two-dimensional and three-dimensional simplexes in Euclidean three-space \(R^{3}\). Prove that the simplex with vertices \(\mathrm{x}_{1}, x_{2}, \ldots, x_{n+1}\) is the set of all points in \(\mathrm{L}\) which can be represented in the form where $$ x=\sum_{k=1}^{n+1} \alpha_{k} x_{k} $$ $$ \alpha_{k} \geqslant 0, \quad \sum^{n+1} \alpha_{k}=1 $$ Problem 4. Show that if the points \(x_{1}, x, \ldots, x_{n+1}\) are in general position, then so are any \(\mathrm{k}+1(\mathrm{k}<\mathrm{n})\) of them. Comment. Hence the \(\mathrm{k}+1\) points generate a k-dimensional simplex, called a \(\mathrm{k}\)-dimensionalface of the \(\mathrm{n}\)-dimensional simplex with vertices \(x_{1}\), \(x_{2}, \ldots, x_{n+1}\)
