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For some $n-$dimensional distribution $cal D$ and a vector $a in mathbbR^n$ can we exactly compute,

$$mathbbE_x sim cal D [ max 0, a^top x ] $$

?

• At least for the Gaussian distribution on $mathbbR^n$ is this known?


• At least for say the uniform distribution on $S^n-1$?

If the distribution $f$ is isotropic (that is, $f = f(r)$), this can be calculated exactly whenever $E(|mathbfx|)$ can using $n$-dimensional spherical coordinates. Let $r$ represent the radial coordinate, $theta$ represent the polar angle, and $Omega$ represent the remaining spherical angles, which form the surface of a unit sphere in $n-1$ dimensions. Without loss of generality, assume $mathbfa$ is parallel to the polar axis. Then $mathbfacdot mathbfx = arcostheta$ and
beginmultline
E(maxmathbfacdotmathbfx,0) = int (mathbfacdotmathbfx)f d^nmathbfx \= int_0^inftyint_0^pileft[int_0^2piright]^n-2maxarcostheta,0 f(r)r^n-1sin^n-2theta ,d^n-2Omega ,dtheta, dr.
endmultline
where $r^n-1sin^n-2theta, d^n-2Omega,dtheta,dr$ is the n-dimensional spherical volume element. Since the factors all depend on one variable, we can separate this integral. Since $costheta < 0$ for $theta > pi/2$, we have
beginmultline
...= aint_0^infty rf(r)r^n-1drint_0^pi/2costhetasin^n-2theta dthetaleft[int_0^2piright]^n-2d^n-2Omega \= frac2api^(n-1)/2(n-1)Gamma((n-1)/2)int_0^infty r^n f(r)dr = fracapi^(n-1)/2Gamma((n+1)/2)int_0^infty r^nf(r)dr
endmultline
where we have used $int_0^pi/2sin^n-2thetacostheta,dtheta = 1/(n-1)$ and the fact that a sphere in $n$ dimensions has surface area $2pi^n/2/Gamma(n/2)$.

Lastly, from
$$
E(|mathbfx|) = int |mathbfx|f d^n mathbfx = 2fracpi^n/2Gamma(n/2)int_0^infty r^nf(r)dr
$$
(again using the $n$-dimensional surface area formula), we have
$$
E(maxmathbfacdotmathbfx,0) =fracapi^(n-1)/2Gamma((n+1)/2)int_0^infty r^nf(r)dr = fraca2sqrtpifracGamma(n/2)Gamma((n+1)/2)E(|mathbfx|)
$$
which is the general form for an isotropic distribution $f$.

$ mathbbE_x sim cal D [ max 0, a^tophspace-0.2em x ] $ can also be calculated exactly whenever $ cal D=mathcalN_n(mu,,Sigma) $ is multivariate normal. In this case $ a^tophspace-0.2emx simmathcalN_1(a^tophspace-0.2emmu,,a^topSigma a) $, so putting $ alpha=a^tophspace-0.2emmu $ and $ sigma=sqrta^topSigma a $, we get
begineqnarray mathbbE_x sim cal D [ max 0, a^tophspace-0.2em x ]&=&frac1sqrt2pisigmaint_-infty^inftymax 0, ye^-fracleft(y-alpharight)^22sigma^2dy\
&=& frac1sqrt2pisigmaint_0^infty ye^-fracleft(y-alpharight)^22sigma^2dy\
&=& frac1sqrt2piint_-fracalphasigma^inftyleft(alpha+sigma zright)e^-fracz^22dz\
&=& alphaleft(1-mathcalN_1(0,,1)left(-fracalphasigmaright)right)+fracsigmasqrt2pi e^-fracalpha^22sigma^2
endeqnarray