Let $(X,\|\|_X)$ be a normed space, $x_1, ..., x_{N} \in X$.Denote $C=[-1, 1]^{N}$ and let $\int_C\|\sum_{i=1}^{N} a_ix_i\|d\mu(a)=1$, where $\mu$ is a Lebesgue measure on $C$. Let $r_i(t), i=1, ..., N$ be a Radamacher functions, i.e. $P(r(t)=1)=P(r(t)=-1)$.
By the fact that $(a_i)$ and $r(t)_ia_i$ have the same distribution and using triangle inequality we can write: $$ \left(\int_C\|\sum_{i=1}^{N} a_ix_i\|^pd\mu(a)\right)^{1/p} =\left(\int_0^1\int_C\|\sum_{i=1}^{N} a_ir_i(t)x_i\|^pd\mu(a)dt\right)^{1/p} \geq \frac 12\left(\int_0^1\|\sum_{i=1}^{N} r_i(t)x_i\|^pdt\right)^{1/p} $$
Assume now the following condition on Rademacher functions: $$ \sum_{i=1}^N r_i(t)=M, \quad 0\leq |M|\leq N. $$ Geometrically this condition would represent a hyper plane $H=\{t:\sum_{i=1}^N r_i(t)=M\}$ intersecting cube $C$. It is know that any slice of $N$-cube perpendicular to main diagonal of the cube may be seen as intersection of two simplices of dimension $(N-1)$ and intersection is the regular polytope $P(N,M)$.
what would be the measure on that polytope and how it will affect above inequality?