Mixing
A variant of submodularity
$begingroup$
A function $f: mathbb{R}^2 to mathbb{R}$ is said to be submodular if for all $x,y in mathbb{R}^2$ it holds
$$
f(x vee y)+f(x wedge y)le f(x)+f(y).
$$
In particular, if $x_1 ge y_1$ and $x_2 le y_2$, this means that
$$
f(y_1,x_2)+f(x_2,y_1) le f(x_1,y_1)+f(x_2,y_2),
$$
or, equivalently,
$$
sum_{Isubseteq {1,2}}(-1)^{|I|}f(xIy)ge 0
$$
where $xIy$ is the vector $I$ where we replace the components of $x$ with the components of $y$ in the positions in $I$.
Question. Let us take a function $f:mathbb{R}^3 to mathbb{R}$ with the property that
$$
sum_{Isubseteq {1,2,3}}(-1)^{|I|}f(xIy)ge 0
$$
for all vector $x,y in mathbb{R}^3$. Do such function have a name in the literature?
reference-request
$endgroup$
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$begingroup$
A function $f: mathbb{R}^2 to mathbb{R}$ is said to be submodular if for all $x,y in mathbb{R}^2$ it holds
$$
f(x vee y)+f(x wedge y)le f(x)+f(y).
$$
In particular, if $x_1 ge y_1$ and $x_2 le y_2$, this means that
$$
f(y_1,x_2)+f(x_2,y_1) le f(x_1,y_1)+f(x_2,y_2),
$$
or, equivalently,
$$
sum_{Isubseteq {1,2}}(-1)^{|I|}f(xIy)ge 0
$$
where $xIy$ is the vector $I$ where we replace the components of $x$ with the components of $y$ in the positions in $I$.
Question. Let us take a function $f:mathbb{R}^3 to mathbb{R}$ with the property that
$$
sum_{Isubseteq {1,2,3}}(-1)^{|I|}f(xIy)ge 0
$$
for all vector $x,y in mathbb{R}^3$. Do such function have a name in the literature?
reference-request
$endgroup$
add a comment |
$begingroup$
A function $f: mathbb{R}^2 to mathbb{R}$ is said to be submodular if for all $x,y in mathbb{R}^2$ it holds
$$
f(x vee y)+f(x wedge y)le f(x)+f(y).
$$
In particular, if $x_1 ge y_1$ and $x_2 le y_2$, this means that
$$
f(y_1,x_2)+f(x_2,y_1) le f(x_1,y_1)+f(x_2,y_2),
$$
or, equivalently,
$$
sum_{Isubseteq {1,2}}(-1)^{|I|}f(xIy)ge 0
$$
where $xIy$ is the vector $I$ where we replace the components of $x$ with the components of $y$ in the positions in $I$.
Question. Let us take a function $f:mathbb{R}^3 to mathbb{R}$ with the property that
$$
sum_{Isubseteq {1,2,3}}(-1)^{|I|}f(xIy)ge 0
$$
for all vector $x,y in mathbb{R}^3$. Do such function have a name in the literature?
reference-request
$endgroup$
A function $f: mathbb{R}^2 to mathbb{R}$ is said to be submodular if for all $x,y in mathbb{R}^2$ it holds
$$
f(x vee y)+f(x wedge y)le f(x)+f(y).
$$
In particular, if $x_1 ge y_1$ and $x_2 le y_2$, this means that
$$
f(y_1,x_2)+f(x_2,y_1) le f(x_1,y_1)+f(x_2,y_2),
$$
or, equivalently,
$$
sum_{Isubseteq {1,2}}(-1)^{|I|}f(xIy)ge 0
$$
where $xIy$ is the vector $I$ where we replace the components of $x$ with the components of $y$ in the positions in $I$.
Question. Let us take a function $f:mathbb{R}^3 to mathbb{R}$ with the property that
$$
sum_{Isubseteq {1,2,3}}(-1)^{|I|}f(xIy)ge 0
$$
for all vector $x,y in mathbb{R}^3$. Do such function have a name in the literature?
reference-request
reference-request
asked Feb 1 at 14:52
user207096
add a comment |
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