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Entropic analogues of Grünbaum's inequality

This paper establishes sharp entropic analogues of Grünbaum's inequality for log-concave random variables, providing bounds on conditional differential entropy in terms of the original entropy and characterizing the equality cases.

Original authors: Matthieu Fradelizi, Lampros Gavalakis, Martin Rapaport

Published 2026-07-28

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Matthieu Fradelizi, Lampros Gavalakis, Martin Rapaport

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to understand the shape of a hidden object, but you can only see it through a foggy window. In the world of mathematics, this "object" is often a cloud of data points, and the "fog" is a concept called entropy. Think of entropy not as a measure of messiness, but as a measure of uncertainty or surprise. If you have a bag of marbles where every single one is red, there is zero surprise when you pull one out; the uncertainty is low. But if the bag has a chaotic mix of red, blue, green, and yellow marbles, pulling one out is a huge surprise; the uncertainty is high.

Now, imagine these marbles aren't just scattered randomly, but they follow a specific rule: they are log-concave. In plain English, this means the marbles are clustered tightly in the middle and taper off smoothly toward the edges, like a perfect hill or a bell curve. Mathematicians have long known a cool trick about these shapes: if you slice the hill in half with a knife that passes through the very center of gravity (the average spot), you are guaranteed to keep at least a specific chunk of the hill on one side. This is a famous rule called Grünbaum's inequality. It's like saying, "No matter how weird your hill looks, as long as it's smooth and centered, you can't slice off more than a certain amount of the 'meat'."

But here is the twist: What if we don't care about the volume of the hill (how much space it takes up), but rather the uncertainty (the entropy) of the data living inside it? Does the same rule apply? If we slice a log-concave hill in half, does the remaining half become more predictable, less predictable, or stay the same? This is the big question the authors of this paper set out to answer. They wanted to know if the "volume" rules of geometry have a "surprise" equivalent in the world of information.

The Great Slice and Uncertainty Hunt

In this paper, the authors act like culinary detectives, taking a smooth, centered hill of data and slicing it with a knife. They ask: "If we chop off the left side of this hill (keeping everything to the right of a cut), does the remaining part become more certain (less surprising) or less certain?"

Their first big discovery is a bit of a relief, but with a very important condition. They prove that for these smooth, centered hills in one dimension, chopping off a tail never makes the remaining part more surprising. In fact, if you slice off the left side (keeping $X \le m$ ), the uncertainty of what's left actually goes down (or stays the same). It's like taking a bag of mixed marbles and removing the weird, rare colors; the bag you have left feels more predictable. They showed this holds true for any "order" of surprise measurement, not just the standard kind. If you have a log-concave distribution in one dimension, cutting off a tail always results in a piece that is at least as "ordered" as the original whole.

However, the story gets more interesting when they flip the question. Instead of asking "Does the piece get smaller in surprise?", they asked, "How much less surprising can it get?" They wanted to find the sharpest possible limit. They knew that if you slice a hill exactly through its center, you can't just say "it gets less surprising." They wanted to know the exact amount of surprise you lose.

Here is where the paper gets its "Aha!" moment. They found that the answer depends entirely on the shape of the hill.

The "Exponential" Champion: If the hill looks like a classic exponential curve (a steep drop-off that flattens out, like a slide), and you slice it right at the center, you lose the maximum amount of surprise possible. The math shows that the uncertainty drops by a very specific, messy-looking number: $\frac{e}{e-1}H_2(1/e)$ . (Don't worry about the math symbols; just know it's a precise constant derived from the number $e$ ). This happens only if the data follows that specific "slide" shape.
The "Flat-Topped" Champion: But wait! If they measure a different kind of surprise called "min-entropy" (which cares mostly about the single most likely spot, the peak of the hill), the winner changes. The shape that loses the most surprise here is a hill that is flat on top for a while and then drops off exponentially. It's like a mesa or a table mountain. For this shape, the uncertainty drops by a different constant: $\log(1 + \sqrt{2})$ .

The authors proved that these are the only two shapes that can reach these limits. If your data looks like anything else, you won't lose as much surprise as these two special cases. It's like finding the two specific keys that unlock the maximum amount of a treasure chest; no other key will turn the lock quite as far.

The High-Dimensional Trap

The paper also tried to see if these rules work in higher dimensions—imagine slicing a 3D ball or a 4D hyper-ball instead of a 2D hill. The authors were hopeful at first, but they hit a wall. They showed that in higher dimensions, the simple rules break down without a correction factor.

They built a counter-example using a cloud of independent data points (like a cloud of 100 separate dice rolls). When they sliced this high-dimensional cloud, they found that the "surprise" of the remaining piece could actually increase as the dimensions got larger, unless you account for the size of the dimension. It turns out that in high dimensions, the geometry gets so weird that the "center slice" doesn't behave like it does in 1D. The authors proved that you can't just copy-paste their 1D formulas to 3D or 100D; you would need to add a "correction factor" that grows with the size of the dimension. They even posed a new question to the math world: "What is the best possible correction factor we can hope for?"

Why This Matters

So, what's the takeaway? The authors have successfully mapped out the relationship between the shape of data and its uncertainty when you cut off a tail. They proved that for smooth, centered data in one dimension, cutting off a tail always reduces uncertainty, and they found the exact "worst-case" scenarios (the exponential and the flat-topped exponential) that define the limits of this reduction.

They didn't just guess; they provided rigorous mathematical proofs, characterizing exactly which shapes achieve these limits. While their rules work perfectly for one-dimensional data, they also showed us that the world gets much more complicated in higher dimensions, where the simple "slice and reduce" logic fails unless you add a dimensional correction. This gives mathematicians a clear boundary: here is where the rules hold, and here is where they break, inviting future explorers to figure out how to fix the rules for the complex, multi-dimensional world we actually live in.

Technical Summary: Entropic Analogues of Grunbaum's Inequality

Problem Statement
The paper investigates the existence of entropic analogues to the classical Grunbaum inequality for convex bodies. The classical inequality asserts that for a convex body $K$ in $\mathbb{R}^n$ , any halfspace $H^+$ containing the barycenter of $K$ cuts off a volume of at least $1/e$ . Its functional counterpart states that for any log-concave random variable $X$ , $P(X \geq \mathbb{E}X) \geq 1/e$ , with equality if and only if $X$ is exponentially distributed.

Motivated by the analogy between entropy and volume, and the growing role of entropy methods in convex geometry, the authors address the question of how the entropy of a log-concave distribution behaves when restricted to a halfspace. Specifically, they seek to establish inequalities relating the entropy of a log-concave random variable $X$ to the entropy of its truncations $X|X \in H^+$ and $X|X \in H^-$ .

Methodology
The authors employ a combination of information-theoretic identities, convex analysis, and optimization techniques:

Information-Theoretic Decomposition: The proof utilizes the identity relating the differential entropy of a variable to the entropies of its conditional distributions and the binary entropy of the truncation probability: $h(X) = p_+ h(X_+) + p_- h(X_-) + H_2(p_+)$ .
Monotonicity and Rigidity: For the upper bound results, the authors establish that Rényi entropy is non-increasing under one-sided truncation of a log-concave density. This relies on a rigidity result for the comparison inequality $\log \|g\|_\infty \leq 1 + \int g \log g$ , characterizing equality cases as specific one-sided or asymmetric double-exponential families.
Constrained Entropy Minimization: For the lower bound (reverse inequality) results, which require the truncation point to be the mean, the authors develop a new constrained entropy minimization result for log-concave densities on $\mathbb{R}^+$ $R^{+}$ .
- They utilize the "degrees of freedom" method to reduce the class of potential minimizers to densities that are log-affine on at most three intervals.
- They apply Fritz–John stationarity conditions (KKT-type optimization) to prove that any local minimizer must actually be log-affine on a single interval (one-piece log-affine).
- This allows for direct verification of the inequality for the resulting exponential or piecewise exponential forms.
Counterexamples: To address higher-dimensional extensions, the authors construct specific counterexamples using product measures and Central Limit Theorem arguments to demonstrate the failure of direct dimensional generalizations without additive dimensional constants.

Key Contributions and Results

1. Upper Bounds (Truncation Inequalities)
The authors prove that for a log-concave random variable $X$ on $\mathbb{R}$ , the Rényi entropy of order $\alpha \in [0, \infty]$ does not increase under one-sided truncation.

Theorem 1: For any $m \in \mathbb{R}$ , $h_\alpha(X | X \leq m) \leq h_\alpha(X)$ .
Equality Characterization: Equality holds if and only if the support of $X$ is contained in $(-\infty, m]$ or $X$ follows an exponential distribution on $(-\infty, M]$ for some $M \geq m$ .
Corollary: This implies that the function $t \mapsto h_\alpha(X | X \leq t)$ is non-decreasing, and $t \mapsto h_\alpha(X | X \geq t)$ is non-increasing.

2. Lower Bounds (Reverse Grunbaum Inequalities)
The authors establish sharp reverse inequalities for differential entropy ( $\alpha=1$ ) and min-entropy ( $\alpha=\infty$ ) under the assumption that the truncation occurs at the mean ( $\mathbb{E}X = 0$ ).

Theorem 2: Let $X$ $X$ be a centered log-concave random variable.
- Differential Entropy: $h(X | X \leq 0) \geq h(X) - C_1$ , where $C_1 = \frac{e}{e-1}H_2(1/e)$ . Equality holds if and only if $X$ follows a centered exponential distribution on $[1/\lambda, \infty)$ for some $\lambda < 0$ .
- Min-Entropy: $h_\infty(X | X \leq 0) \geq h_\infty(X) - C_\infty$ , where $C_\infty = \log(1+\sqrt{2})$ . Equality holds if and only if the density is constant on a finite interval $[0, a]$ and exponential on $(-\infty, 0]$ .
Significance of Extremizers: The extremizers differ by entropy order. The differential entropy extremizer is a mirrored exponential, while the min-entropy extremizer is a piecewise log-affine function (constant on one side, exponential on the other).

3. Higher-Dimensional Obstructions
The paper demonstrates that these results do not extend directly to $\mathbb{R}^n$ without modification:

Centering Necessity: A counterexample in dimension 2 shows that without the truncation hyperplane passing through the barycenter, the min-entropy inequality fails.
Dimensional Constants: Using a Central Limit Theorem argument on product exponentials, the authors show that for large $n$ , the difference $h(X_+) - h(X)$ grows as $\sqrt{n}$ . Thus, any dimensional extension of the reverse inequality requires an additive constant $C_n$ of order $\Omega(\sqrt{n})$ .

Significance and Claims
The paper claims to provide the first entropic analogues of Grunbaum's inequality with complete characterizations of equality cases.

Sharpness: The constants $C_1$ and $C_\infty$ are proven to be sharp, and the specific distributions achieving equality are fully identified.
Theoretical Bridge: The work connects the geometric Grunbaum inequality with information theory, reinforcing the analogy between volume and entropy.
New Inequalities: The proof of the lower bound relies on a new inequality (Proposition 17) regarding the entropy of log-concave densities on $\mathbb{R}^+$ with fixed mean and boundary slope constraints. The authors note that the characterization of equality cases in the comparison between differential and min-entropy (Proposition 10) may be of independent interest.
Reliability Connection: The authors observe that their results imply log-concave distributions in one dimension possess "decreasing uncertainty of residual life," a property previously studied in reliability engineering.

The paper concludes by posing open questions regarding the optimal constants $c_n, C_n$ for higher dimensions and the behavior of entropy under conditioning on hyperplanes with different normal vectors, suggesting that product exponential densities may be the relevant extremizers in higher dimensions.

The Great Slice and Uncertainty Hunt

The High-Dimensional Trap

Why This Matters

Technical Summary: Entropic Analogues of Grunbaum's Inequality

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