Sharp one-sided concentration conjecture for standardized log-concave distributions

Let $L$ denote the set of all real random variables $X$ with a log-concave probability density function such that $EX=0$ and $EX^2=1$ . Let $Y$ have the standard exponential distribution, so that $Y-1\in L$ . Sharp one-sided concentration conjecture. For every real $\delta>0$ ,

\min_{X\in L}P(0<X<\delta)=P(0<Y-1<\delta)=e^{-1}(1-e^{-\delta}),

and consequently

\min_{X\in L}f_X(0)=e^{-1}.

The conjecture proposes that the centered standard exponential distribution gives the exact extremal lower bound for one-sided concentration near the mean, improving the nonoptimal bound proved earlier in the paper. The preceding discussion explicitly notes that the established bounds are likely far from optimal, while this sharp form remains unproved.

Progress summary

Solved

A reader-posted uniform example claims the conjecture is false, but the counterexample has not been independently verified.

The conjecture says that the centered, variance-one exponential distribution gives the smallest one-sided mass near the mean, and the smallest density at the mean, among standardized log-concave distributions. The proposed extremal formulas remain unsupported by the published source.

Known results

Pinelis (2026) obtained a non-sharp lower bound for $P(0<X<\delta)$ for every $\delta>0$ and every standardized log-concave $X$ ; the abstract does not claim the conjectured constant.
A MathOverflow argument gives only a universal $c>0$ with $P(X\in(-1/2,0))\ge c$ , without identifying the sharp constant or extremizer.

Posted attempt

A reader-posted calculation uses $X\sim\operatorname{Unif}[-\sqrt{3},\sqrt{3}]$ and claims both formulas fail: $f_X(0)=1/(2\sqrt{3})<e^{-1}$ and, at $\delta=1/10$ , the one-sided mass is below the conjectured exponential value. This is a claimed complete disproof, not independently verified.

Current status (as of August 2026): The sharp conjecture is not established, while a posted uniform-distribution counterexample claims to disprove both assertions and remains unverified.

Sources

Sources & referencesView supporting material

Primary source

Iosif Pinelis, “One-sided concentration near the mean of log-concave distributions”, arXiv:2602.06804 (2026).

concentration-inequalities log-concave-distributions probability-inequalities

1 week ago

Both asserted sharp minima fail for the same standardized log-concave distribution.

Let

X\sim\operatorname{Unif}[-\sqrt3,\sqrt3].

Its density

f_X(x)=\frac1{2\sqrt3}\mathbf1_{[-\sqrt3,\sqrt3]}(x)

is log-concave because its support is an interval and the density is constant there. Moreover,

\mathbb E[X]=0,\qquad \mathbb E[X^2]=\frac13(\sqrt3)^2=1,

so $X$ belongs to the exact class in the conjecture.

At the origin,

f_X(0)=\frac1{2\sqrt3}<\frac1e.

Indeed $e<3<2\sqrt3$ . Therefore

\inf_{X\in\mathcal L}f_X(0) \le\frac1{2\sqrt3} < e^{-1},

contradicting the asserted density minimum.

The one-sided concentration claim also fails explicitly. Set

\delta=\frac1{10}.

Then

\mathbb P(0<X<\delta) = \frac1{20\sqrt3}.

On the other hand,

e^{1/10}>1+\frac1{10}=\frac{11}{10},

and hence

1-e^{-1/10}>\frac1{11}.

Using $e<3$ , it follows that

e^{-1}(1-e^{-1/10}) > \frac1{11e} > \frac1{33}.

Finally,

20\sqrt3>33

because $1200>1089$ . Therefore

\mathbb P(0<X<1/10) = \frac1{20\sqrt3} < \frac1{33} < e^{-1}(1-e^{-1/10}).

Thus the centered exponential distribution is not the minimizer proposed in either assertion.

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Shivam Patel · 2 days ago