On the \(P\)-property of convex compact sets. (Q1810078)

If \(A\subset\mathbb{R}^n\) is a compact convex set and \(PA\) is its projection on \((n- 1)\)-dimensional subspace, then the function \(f: PA\to\mathbb{R}\) is defined by the formula \(f(x)= \min\{ y: (x,y)\in A\}\). We say that a compact convex set \(A\subset\mathbb{R}^n\) has the property \(P\) if for each \((n- 1)\)-dimensional subspace the corresponding function \(f\) is continuous on \(PA\). The paper contains a series of properties of compact convex sets having the property \(P\). The main result is Theorem 2.1: If \(A_1\), \(A_2\) have the property \(P\), then the arithmetic sum \(A_1+ A_2\) also has this property. It is shown also that if \(A\) has the property \(P\) and \(T: \mathbb{R}^n\to \mathbb{R}^m\) is a linear operator, then \(TA\) has property \(P\). The relation between generating sets and sets with the property \(P\) is discussed.