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 upper bounded and 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 $m_j:=m_j(x_1,\ldots,x_n):=\max_{t\in [x_j,x_{j+1}]}F(x_1,\ldots,x_n,t)$ ($j=0,1,\ldots,n$). 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.