Maximizing a hyperplane $\sum_i a_ix_i$ where $a_i\in\mathbb R$ and each $a_i$ are fixed and non-negative and $x_i$ are variables over a standard simplex $\sum_i x_i\leq 1$ with $0\leq x_i$ always produces a vertex point on the simplex and maximization corresponds to $\max_i a_i$.

>1. In infinite dimensions is such a proof considered constructive or does it hold only in classical logic? It seems that we would have to show that at maximizing point there is an $i\in\mathbb N$ such that $x_i=1$ holds and for that perhaps is it possible we cannot do this without invoking [LLPO](https://en.wikipedia.org/wiki/Limited_principle_of_omniscience)?

Suppose we are looking for a $0/1$ vector in integers (not reals as in 1.) on the standard simplex and we know the optimal vector has either sum of even coordinates summing to $1$ or odd coordinates summing to $1$ then in finite dimensions it is a process of enumerating vertices.

>2. In infinite dimensions is such a proof considered constructive or does it hold only in classical logic? It seems that we would have to show that at optimizing point there is an $i\in\mathbb N$ such that $x_i=1$ holds and for that perhaps is it possible we cannot do this without invoking [LLPO](https://en.wikipedia.org/wiki/Limited_principle_of_omniscience)?

In general is the proofs of optimization over infinite dimensions considered constructive?

3a. How about when each $a_i$ are fixed and positive?

3b. How about when each $a_i$ are fixed and distinct and non-negative thus guaranteeing an unique vertex point?

3c. How about when each $a_i$ are fixed and distinct and positive thus guaranteeing an unique vertex point?