There is a systematic approach to this typo of problems which finds the best possible constants (when $n\to \infty$) but requires plenty (but easy) computations. I will mention the steps and if somebody is interested I can provide more details

What are we looking for? $$ \sup_{x,y \in L^{p}}\{ \|x+y\|_{L^{p}} : \|x\| _{L^p}=1, \|y\| _{L^p}=1, \| x-y\| _{L^{p}}=1 \} $$

(I am sorry I cannot put brackets {}.) Let us consider the following extremal problem in $L^{p}([0,1])$ $$ B(u,v,w) = \sup_{f,g \in L^{p}} \{ \int_{0}^{1}|f+g|^{p} : \int_{0}^{1}|f|^{p}=u, \int_{0}^{1}|g|^{p}=v, \int_{0}^{1}|f-g|^{p}=w \} $$

Then clearly the best possible $C_{p}=(B(1,1,1))^{1/p}$.

Properties of B

Now the advantage of considering the function $B$ is that it satisfies the following properties

1. $B$ is given in the convex cone $\Omega$ such that $(u,v,w)\in \Omega$ iff $w^{1/p}\leq u^{1/p}+v^{1/p}$, $u^{1/p}\leq v^{1/p}+w^{1/p}$ and $v^{1/p}\leq u^{1/p}+w^{1/p}$.

2. $B$ is a concave function in $\Omega$.

3. $B$ has a boundary condition in $\Omega$ i.e, $B(|x|^{p}, |y|^{p}, |x-y|^{p})=|x+y|^{p}$

4. $B$ is minimal among those who satisfy properties 1), 2) and 3).

Now it is clear how to find $B$: $B$ is a minimal concave function in $\Omega$ with a given boundary conditions 3). Notice that $B$ is 1-homogeneous so it is enough to find $B$ just in any section of the convex cone $\Omega$ (say for example $w=1$).

Your initial question reduced to a purely geometrical question: find the concave envelope of the boundary data.

After you find $B$ you can use it for your finite dimensional problem: if $d\mu$ is the uniform counting measure with weights $1/n$ then by Jensen's inequality

$$ \int |x+y|^{p} d\mu = \int B(|x|^{p},|y|^{p},|x-y|^{p})d\mu \leq B\left(\int |x|^{p} d\mu, \int |y|^{p}, \int |x-y|^{p} \right) $$ 1-homogeneity of $B$ and the right hand side gives you an upper bound.

There is a systematic approach to these type of problems which finds the best possible constants (when $n\to \infty$) but requires plenty (but easy) computations. I will mention the steps and if somebody is interested I can provide more details

What are we looking for? $$ \sup_{x,y \in L^{p}}\{ \|x+y\|_{L^{p}} : \|x\| _{L^p}=1, \|y\| _{L^p}=1, \| x-y\| _{L^{p}}=1 \} $$

Let us consider the following extremal problem in $L^{p}([0,1])$ $$ B(u,v,w) = \sup_{f,g \in L^{p}} \{ \int_{0}^{1}|f+g|^{p} : \int_{0}^{1}|f|^{p}=u, \int_{0}^{1}|g|^{p}=v, \int_{0}^{1}|f-g|^{p}=w \} $$

Then clearly the best possible $C_{p}=(B(1,1,1))^{1/p}$.

Properties of B

Now the advantage of considering the function $B$ is that it satisfies the following properties

1. $B$ is given in the convex cone $\Omega$ such that $(u,v,w)\in \Omega$ iff $w^{1/p}\leq u^{1/p}+v^{1/p}$, $u^{1/p}\leq v^{1/p}+w^{1/p}$ and $v^{1/p}\leq u^{1/p}+w^{1/p}$.

2. $B$ is a concave function in $\Omega$.

3. $B$ has a boundary condition in $\Omega$ i.e, $B(|x|^{p}, |y|^{p}, |x-y|^{p})=|x+y|^{p}$

4. $B$ is minimal among those who satisfy properties 1), 2) and 3).

Now it is clear how to find $B$: $B$ is a minimal concave function in $\Omega$ with a given boundary conditions 3). Notice that $B$ is 1-homogeneous so it is enough to find $B$ just in any section of the convex cone $\Omega$ (say for example $w=1$).

Your initial question reduced to a purely geometrical question: find the concave envelope of the boundary data.

After you find $B$ you can use it for your finite dimensional problem: if $d\mu$ is the uniform counting measure with weights $1/n$ then by Jensen's inequality

$$ \int |x+y|^{p} d\mu = \int B(|x|^{p},|y|^{p},|x-y|^{p})d\mu \leq B\left(\int |x|^{p} d\mu, \int |y|^{p}, \int |x-y|^{p} \right) $$ 1-homogeneity of $B$ and the right hand side gives you an upper bound.

There is a systematic approach to this typo of problems which finds the best possible constants (when $n\to \infty$) but requires plenty (but easy) computations. I will mention the steps and if somebody is interested I can provide more details

What are we looking for? $$ \sup_{x,y \in L^{p}}\{ \|x+y\|_{L^{p}} : \|x\| _{L^p}=1, \|y\| _{L^p}=1, \| x-y\| _{L^{p}}=1 \} $$

(I am sorry I cannot put brackets {}.) Let us consider the following extremal problem in $L^{p}([0,1])$ $$ B(u,v,w) = \sup_{f,g \in L^{p}} \{ \int_{0}^{1}|f+g|^{p} : \int_{0}^{1}|f|^{p}=u, \int_{0}^{1}|g|^{p}=v, \int_{0}^{1}|f-g|^{p}=w \} $$

Then clearly the best possible $C_{p}=B(1,1,1)$.

Properties of B

Now the advantage of considering the function $B$ is that it satisfies the following properties

1. $B$ is given in the convex cone $\Omega$ such that $(u,v,w)\in \Omega$ iff $w^{1/p}\leq u^{1/p}+v^{1/p}$, $u^{1/p}\leq v^{1/p}+w^{1/p}$ and $v^{1/p}\leq u^{1/p}+w^{1/p}$.

2. $B$ is a concave function in $\Omega$.

3. $B$ has a boundary condition in $\Omega$ i.e, $B(|x|^{p}, |y|^{p}, |x-y|^{p})=|x+y|^{p}$

4. $B$ is minimal among those who satisfy properties 1), 2) and 3).

Now it is clear how to find $B$: $B$ is a minimal concave function in $\Omega$ with a given boundary conditions 3). Notice that $B$ is 1-homogeneous so it is enough to find $B$ just in any section of the convex cone $\Omega$ (say for example $w=1$).

Your initial question reduced to purely geometrical questions: find the concave envelope of the boundary data.

After you find $B$ you can use it for your finite dimensional problem: if $d\mu$ is the uniform counting measure with weights $1/n$ then by Jensen's inequality

$$ \int |x+y|^{p} d\mu = \int B(|x|^{p},|y|^{p},|x-y|^{p})d\mu \leq B\left(\int |x|^{p} d\mu, \int |y|^{p}, \int |x-y|^{p} \right) $$ 1-homogeneity of $B$ and the right hand side gives you an upper bound.

There is a systematic approach to this typo of problems which finds the best possible constants (when $n\to \infty$) but requires plenty (but easy) computations. I will mention the steps and if somebody is interested I can provide more details

What are we looking for? $$ \sup_{x,y \in L^{p}}\{ \|x+y\|_{L^{p}} : \|x\| _{L^p}=1, \|y\| _{L^p}=1, \| x-y\| _{L^{p}}=1 \} $$

(I am sorry I cannot put brackets {}.) Let us consider the following extremal problem in $L^{p}([0,1])$ $$ B(u,v,w) = \sup_{f,g \in L^{p}} \{ \int_{0}^{1}|f+g|^{p} : \int_{0}^{1}|f|^{p}=u, \int_{0}^{1}|g|^{p}=v, \int_{0}^{1}|f-g|^{p}=w \} $$

Then clearly the best possible $C_{p}=(B(1,1,1))^{1/p}$.

Properties of B

Now the advantage of considering the function $B$ is that it satisfies the following properties

1. $B$ is given in the convex cone $\Omega$ such that $(u,v,w)\in \Omega$ iff $w^{1/p}\leq u^{1/p}+v^{1/p}$, $u^{1/p}\leq v^{1/p}+w^{1/p}$ and $v^{1/p}\leq u^{1/p}+w^{1/p}$.

2. $B$ is a concave function in $\Omega$.

3. $B$ has a boundary condition in $\Omega$ i.e, $B(|x|^{p}, |y|^{p}, |x-y|^{p})=|x+y|^{p}$

4. $B$ is minimal among those who satisfy properties 1), 2) and 3).

Now it is clear how to find $B$: $B$ is a minimal concave function in $\Omega$ with a given boundary conditions 3). Notice that $B$ is 1-homogeneous so it is enough to find $B$ just in any section of the convex cone $\Omega$ (say for example $w=1$).

Your initial question reduced to a purely geometrical question: find the concave envelope of the boundary data.

After you find $B$ you can use it for your finite dimensional problem: if $d\mu$ is the uniform counting measure with weights $1/n$ then by Jensen's inequality

$$ \int |x+y|^{p} d\mu = \int B(|x|^{p},|y|^{p},|x-y|^{p})d\mu \leq B\left(\int |x|^{p} d\mu, \int |y|^{p}, \int |x-y|^{p} \right) $$ 1-homogeneity of $B$ and the right hand side gives you an upper bound.