\(p\)-adic convex functions (Q1970254)

Let \(K\) be a field with a non-trivial non-Archimedean valuation \(|\cdot |\), \(C\) be a \(K\)-convex set containing at least two points. A function \(f:\;C\to [0,\infty)\) is called \(K\)-convex if for \(x,y\in C\), \(\lambda ,\mu \in K\), \(|\lambda |\leq 1\), \(\lambda +\mu =1\) we have \[ f(\lambda x+\mu y)\leq \max (|\lambda |f(x),|\mu |f(y)). \] The author studies some properties of \(K\)-convex functions. A characterization of a \(K\)-convex function as a maximum of certain simple functions is presented. Another characterization gives an expression of a \(K\)-convex function in terms of some increasing radial function. An extension theorem for \(K\)-convex functions is also proved.