# Is there a function whose maximizers remain the same after any affine transformations?

Let $$f: \mathbb{R_+}^n\to \mathbb{R_+}$$ be a function that is strictly increasing in each of its arguments. Let $$M_f$$ be the set of its maximizers on some fixed compact subset $$D\subseteq \mathbb{R_+}^n$$, that is, $$M_f := \arg \max_{x\in D} f(x)$$.

Let $$g$$ be a function on $$\mathbb{R}^n$$ that multiplies each coordinate $$i$$ by a positive constant $$a_i$$, so that $$g(x_1,\ldots,x_n) = (a_1 x_1,\ldots , a_n x_n)$$. In general, the set of maximizers of $$f\circ g$$ is different than the set of maximizers of $$f$$. For example, if $$f$$ is the sum function, $$f(x_1,\ldots,x_n) = x_1+\cdots+x_n$$, then $$(f\circ g)(x_1,\ldots,x_n) = a_1 x_1+\cdots+a_n x_n$$, and the set of maximizers is clearly different. However, if $$f$$ is the product function, $$f(x_1,\ldots,x_n) = x_1\cdot \cdots \cdot x_n$$, then $$(f\circ g)(x_1,\ldots,x_n) = (a_1\cdots a_n)\cdot (x_1\cdots x_n)$$, and because all constants $$a_i$$ are positive, the set of maximizers of $$f$$ and $$f\circ g$$ is exactly the same.

Now, let $$h$$ be a function on $$\mathbb{R}^n$$ that adds a constant $$b_i$$ to each coordinate, so that $$h(x_1,\ldots,x_n) = (x_1+b_1,\ldots , x_n+b_n)$$. In this case, the set of maximizers of $$f\circ h$$ is equal to the set of maximizers of $$f$$ if $$f$$ is the sum function, but not if $$f$$ is the product function.

Finally, let $$z$$ be an affine transformation on $$\mathbb{R}^n$$, $$z(x_1,\ldots,x_n) = (a_1 x_1+b_1,\ldots , a_n x_n+b_n)$$, where $$a_i$$ are positive constants and $$b_i$$ are constants. Is there a function $$f$$ such that the set of maximizers of $$f$$ is equal to the set of maximizers of $$f\circ z$$, for any constants $$a_i$$ and $$b_i$$? The sum and product functions do not work, but maybe there is another function $$f$$ with this property?

• Related question: math.stackexchange.com/q/4945643/29780 Commented Jul 14 at 19:56
• What is the compact set $D$ here? Arbitrary but fixed? Can $f$ depend on it? Commented Jul 16 at 13:55