$\def\R{\mathbb{R}}$I have been looking for strongly asymmetric functions that are strictly Schur-convex, where the relevant concepts are defined below:

Definition 1: For function $f: \R^n → \R$ and permutation $π$ on $\{1, 2, \cdots, n\}$, if$$ f(x_1, \cdots, x_n) = f(x_{π(1)}, \cdots, x_{π(n)}), \quad \forall x_1, \cdots, x_n \in \R $$ then $f$ is said to be $π$-symmetric. If $f$ is not $π$-symmetric for any non-identity permutation $π$, then $f$ is said to be strongly asymmetric.

Definition 2: Any function $f: \R^n → \R$ that satisfies the following condition is said to be strictly Schur-convex: For any $x, y \in \R^n$, if $x_i > y_i \geqslant y_j > x_j$, $x_i + x_j = y_i + y_j$ and $x_k = y_k$ ($\forall k ≠ i, j$) for some $1 \leqslant i, j \leqslant n$, then $f(x) > f(y)$.

Remark: The above definition of Schur-convexity is different from that in [1], where it requires the symmetry of $f$.

Current progress

For $n = 2$, I have found the following example:$$ f_1(x_1, x_2) = |x_1 - x_2| + x_2. $$ To see that $f_1$ is strictly Schur-convex, define $g_1(h; x) = f_1(x + h, x - h)$ for $x, h \in \R$. Since$$ g_1(h; x) = 2|h| + (x - h) = \begin{cases} x - 3h; & h \leqslant 0\\ x + h; & h > 0 \end{cases}, $$ then $g_1(h; x)$ is strictly decreasing for $h < 0$ and strictly increasing for $h \geqslant 0$. Therefore $f_1$ is strictly Schur-convex.

Next, my intuition was to generalize this construction for $n = 3$ as$$ f_2(x_1, x_2, x_3) = |x_1 - x_2| + |x_1 - x_3| + a_2 x_2 + a_3 x_3, $$ where $a_2$ and $a_3$ are distinct constants to be chosen. Analogously defining $g_2(h; x, x_1) = f_2(x_1, x + h, x - h)$ for $x, x_1, h \in \R$, then\begin{align*} g_2(h; x, x_1) &= |x_1 - x - h| + |x_1 - x + h| + (a_2 - a_3) h + (a_2 + a_3) x\\ &= \begin{cases} 2|x_1 - x| + (a_2 - a_3) h + (a_2 + a_3) x; & |h| \leqslant |x_1 - x|\\ 2|h| + (a_2 - a_3) h + (a_2 + a_3) x; & |h| > |x_1 - x| \end{cases}. \end{align*} Schur-convexity requires that $g_2$ be strictly decreasing for $h \leqslant 0$ and strictly increasing for $h > 0$, but considering $g_2(h; x, x_1)$ for $|h| \leqslant |x_1 - x|$ leads to a contradiction. So this might not be a correct way to find examples for $n \geqslant 3$.

Are there any asymmetric functions with explicit expressions that are strictly Schur-convex for $n \geqslant 3$? It would be better if such examples are sums of linear functions and/or their absolute values. Thanks in advance.

  • $\begingroup$ What's the motivation for Definition 2? BTW, in the book, the symmetry is not required, it's a consequence of the properties of the order $\prec$ on $\mathbb{R}^n$. $\endgroup$ – user742759 Jun 23 at 16:24
  • $\begingroup$ Also if you show your work how you found that example, that would save time for people trying to help you. $\endgroup$ – user742759 Jun 23 at 16:27
  • $\begingroup$ @yumi Symmetry is implied by majorization but not by strict majorization, and definition 2 rewrites A.2.b in chapter 3, part I of the book. $\endgroup$ – Saad Jun 24 at 0:16
  • $\begingroup$ @yumi The construction of $f_2$, sort of by chance, is by adding extra terms to $|x_1-x_2|$ to make $g_1$ increase and derease in respective regions. $\endgroup$ – Saad Jun 24 at 0:18
  • $\begingroup$ To downvoters: why not spend some time reading How to ask a good question and improve your own posts instead of casting malicious down votes here? $\endgroup$ – Saad Jun 24 at 1:19

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