A strengthened truncated-mean inequality for sums of uniform random variables

The conjecture holds for every $n\ge1$ and every $-1<t<1$. In fact, there is an exact positive-deficit identity, and the constant $1/(2n)$ is sharp.

Write

$$X=\frac1n\sum_{i=1}^nU_i,\qquad m=\mathbb E[X\mid X>t],$$

and define

$$V=\operatorname{Var}(X\mid X>t), \qquad D= \mathbb E\left[ \frac1n\sum_{i=1}^n(U_i-X)^2 \,\middle|\,X>t \right].$$

We prove

$$\boxed{\displaystyle m(m-t) = \frac{1-m^2}{2n} - \left(1+\frac1{2n}\right)V - \frac D{2n}.} \tag{1}$$

For $U\sim\operatorname{Unif}[-1,1]$, integration by parts gives

$$\mathbb E[Uh(U)] = \mathbb E\left[ \frac{1-U^2}{2}h'(U) \right],$$

since the boundary term vanishes at $U=\pm1$. Apply this conditionally in coordinate $U_i$ to

$$h(U_i)=(X-t)_+, \qquad h'(U_i)=\frac1n\mathbf1_{\{X>t\}}$$

almost everywhere. Summing over the independent coordinates yields

$$\mathbb E[X(X-t)_+] = \frac1{2n^2} \mathbb E\left[ \sum_{i=1}^n(1-U_i^2) \mathbf1_{\{X>t\}} \right]. \tag{2}$$

Divide by $\mathbb P(X>t)>0$. The left side becomes

$$\mathbb E[X^2\mid X>t]-tm =V+m(m-t).$$

Meanwhile,

$$\frac1n\sum_{i=1}^nU_i^2 = X^2+\frac1n\sum_{i=1}^n(U_i-X)^2,$$

so the right side of (2), after conditioning, becomes

$$\frac{1-m^2-V-D}{2n}.$$

Equating these expressions proves (1).

Every nonempty upper tail has strictly positive conditional variance $V>0$, while $D\ge0$ and $0<m<1$. Therefore

$$\boxed{\displaystyle \frac{m(m-t)}{1-m^2} = \frac1{2n} - \frac{(2n+1)V+D}{2n(1-m^2)} < \frac1{2n}.}$$

This is precisely the conjecture, with an explicit strictness gap.

Finally, if $1-2/n\le t<1$ and $h=1-t$, the coordinate deficits $1-U_i$, conditioned on $X>t$, are uniformly distributed on

$$\left\{y_i\ge0:\sum_{i=1}^ny_i<nh\right\}.$$

Hence

$$m=1-\frac{nh}{n+1}$$

and

$$\frac{m(m-t)}{1-m^2} = \frac{m}{n(1+m)} \longrightarrow\frac1{2n} \qquad(t\uparrow1).$$

Thus the conjectured constant is best possible for every $n$.

Source: S. Gilboa, P. Haim-Kislev and B. Slomka, Isobarycentric Inequalities, International Mathematics Research Notices 2023(14), 12298–12323, doi:10.1093/imrn/rnac191; arXiv:2202.07527, Conjecture 1.5.