A homeomorphism theorem for sums of translates

For a fixed positive integer $n$ consider continuous functions $K_1,\dots$, $ K_n:[-1,1]\to \mathbb{R}\cup\{-\infty\}$ that are concave and real valued on $[-1,0)$ and on $(0,1]$, and satisfy $K_j(0)=-\infty$. Moreover, let $J:[0,1]\to \mathbb{R}\cup\{-\infty\}$ be such that $[0,1]\setminus J^{-1}(\{-\infty\})$ has at least $n+1$ elements, but it is arbitrary otherwise. For $x_0:=0<x_1<\dots< x_n \le x_{n+1}:=1$, so called nodes, and for $t\in [0,1]$ consider the sum of translates function' $F(x_1,\ldots,x_n,t):=J(t)+\sum_{j=1}^n K_j(t-x_j)$, and the vector of interval maximum values $m_j:=m_j(x_1,\ldots,x_n):=\max_{t\in [x_j,x_{j+1}]} F$ ($j=0,1,\ldots,n$). We describe the structure of the arising interval maxima as the nodes run over the $n$-dimensional simplex. Applications presented here range from abstract moving node Hermite--Fej\'er interpolation for generalized algebraic and trigonometric polynomials via Bojanov's problem to more abstract results of interpolation theoretic flavour. Finally, more subtle applications concerning diverse minimax, equioscillation, maximin existence and characterization results, are previewed, whose detailed description follows in a forthcoming, companion paper.