Minimum value of the maximal entry of n-dimensional vectors with constant sum of the entries and of the squared entries

We present the analytical study of a constrained non-linear optimization problem relevant to the optimization of fractionated radiotherapy protocols of length $n$. The admissible set considered here is derived from the imposition of constraints, mandatory in radiotherapy, to guarantee that the damages caused to healthy tissues by the radiation do not exceed assigned tolerable levels. Radiation damages are evaluated by means of the well-known LQ model and, for suitable values of the normal tissue parameters, the constraints are written as a linear constraint and a quadratic constraint. In this report, we prove a property satisfied by the value of the maximal entry of vectors in the mentioned feasible region, and precisely we determine the minimal value of the maximal entry of such vectors. This result is significant for the problem of radiotherapy optimization when an upper bound is set on the size of the daily fraction doses in addition to the normal tissue constraints [3]. Indeed, the optimum of the present problem acts as a threshold with regard to the dose upper bound influencing the type of solution of the radiotherapy optimization problem.