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On non-symmetric $t$ -convexity

This paper resolves a question posed by Zsolt Páles by proving that any function $f: \mathbb{R}\to \mathbb{R}$ satisfying the inequality $f\left(\frac{x+2y}{3}\right) \le \frac{f(x)+2f(y)}{3}$ for all $x\le y$ must necessarily be midconvex.

Original authors: Paolo Leonetti

Published 2026-08-04

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: Paolo Leonetti

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 a world where the rules of "bending" are slightly different depending on which way you look. In the branch of mathematics known as functional analysis, scientists study functions—think of them as machines that take a number in and spit a number out. A classic rule for these machines is called "convexity." If you picture a function as a curve, a convex curve is like a smiley face or a bowl; it never dips below a straight line drawn between any two points on it. This is the "midpoint" rule: if you pick two points, the spot exactly in the middle of the curve must be lower than (or equal to) the spot exactly in the middle of the straight line connecting them.

But what if we change the rules? What if we don't check the exact middle, but instead check a spot that is one-third of the way from one side and two-thirds from the other? This is called "t-convexity," where "t" is just a fancy letter for that specific fraction. Now, here is the twist: usually, math rules apply no matter which order you put your numbers in. But in this specific corner of math, researchers asked a tricky question: What if the rule only has to hold when you put the smaller number first and the larger number second? This is "non-symmetric" convexity. It's like saying a bowl must look like a bowl only if you look at it from left to right, but maybe it can be weird if you look from right to left. For a long time, mathematicians wondered if a function could follow this "one-way" rule without actually being a true, two-way convex curve.

In this paper, mathematician Paolo Leonetti tackles a specific version of this puzzle. He focuses on the case where the "t" value is exactly 1/3. The question, originally posed by Zsolt Páles, was simple to ask but hard to answer: Does there exist a function that follows the "one-way" rule for the 1/3 spot, but fails the "two-way" rule? In other words, can a function be "non-symmetric 1/3-convex" without being "1/3-convex" (which, in this specific case, is the same as being "midconvex")?

Leonetti's answer is a definitive "no." He proves that if a function obeys the one-way rule for the 1/3 mark, it is forced by the laws of mathematics to obey the two-way rule as well. There is no loophole. The function cannot deviate.

To understand how he proved this, imagine the function as a flexible rubber band stretched between two points. The "non-symmetric" rule says that if you pull the band at the 1/3 mark, it must stay below a certain line, but only if you pull from the left side toward the right. Leonetti showed that if you try to build a rubber band that follows this rule but bends the wrong way when you look at it from the other side, the math simply breaks. The tension in the band becomes impossible to sustain.

The proof works like a domino effect. Leonetti starts by assuming such a "deviating" function exists. He then uses a clever mathematical trick to stretch and shrink the function, creating a sequence of points that get closer and closer to zero. He shows that if the function were deviating, the values at these points would have to grow infinitely large, faster than any real number can. It's like trying to fill a bucket with water where the bucket gets smaller every time you pour, but the water volume doubles. Eventually, the bucket would have to hold an infinite amount of water, which is impossible. Since the assumption leads to an impossibility, the "deviating" function cannot exist.

The paper doesn't just stop at 1/3. It suggests that this "no deviation" rule holds true for other fractions too, like 1/4, 1/5, and 1/6. In fact, the author shows that the set of numbers where this rule holds is "dense," meaning if you pick any random spot between 0 and 1, you can find a number very close to it where the rule works. However, there is still a mystery left unsolved: the paper leaves open the question of whether this holds true for all algebraic numbers (a specific type of number involving roots and fractions), or if there are still some hidden, tricky numbers where a deviating function might hide. For now, though, for the specific case of 1/3, the door is closed: non-symmetric convexity is just regular convexity in disguise.

Technical Summary: On Non-Symmetric t-Convexity

Problem Statement
The paper addresses a specific question regarding the relationship between non-symmetric $t$ -convexity and standard $t$ -convexity for real-valued functions. A function $f: \mathbb{R} \to \mathbb{R}$ is defined as non-symmetric $t$ -convex if the inequality
$f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)$
holds for all $x \leq y$ . If the inequality holds for all $x, y \in \mathbb{R}$ , the function is simply $t$ -convex.

It is a known result that for a fixed $t \in (0, 1)$ , non-symmetric $t$ -affine functions are necessarily $t$ -affine. However, Lewicki and Olbryś demonstrated that this implication fails for convexity when $t$ is a transcendental number; they constructed a non-symmetric $t$ -convex function that is not $t$ -convex. This leaves open the question for specific rational values of $t$ . Specifically, Zsolt Páles asked whether there exists a non-symmetric $t$ -convex function that is not $t$ -convex when $t = 1/3$ . Since $1/3$ -convexity is equivalent to midconvexity (1/2-convexity) in the context of rational parameters, the question is whether non-symmetric $1/3$ -convexity implies midconvexity.

Methodology
The author, Paolo Leonetti, employs a functional inequality approach to establish a general criterion for the equivalence of non-symmetric and symmetric $t$ -convexity.

General Criterion (Proposition 2.1): The core of the proof is a proposition stating that for a fixed $t \in (0, 1/2)$ , if there exist a constant $C > 1-t$ , a finite set of ordered pairs $\{(u_i, v_i)\}$ with $u_i \leq v_i$ , non-negative coefficients $\mu_i$ , and a real number $\eta$ such that a specific linear combination of the "defect" functionals $J_{t,\phi}$ equals a specific linear combination of function values, then every non-symmetric $t$ -convex function is $t$ -convex.
The defect functional is defined as $J_{t,\phi}(u, v) = t\phi(u) + (1-t)\phi(v) - \phi(tu + (1-t)v)$ .
Proof by Contradiction and Iteration: Assuming the existence of a non-symmetric $t$ -convex function $f$ that is not $t$ -convex, the author constructs an auxiliary function $F$ (via affine transformation) such that $F(0)=F(1)=0$ and $F(t) > 0$ . By applying the general criterion to scaled versions of this function, the author derives a recurrence relation involving the sequence $G_n = f(t^{n+1}) - tf(t^n)$ .
The proof demonstrates that under the condition $C > 1-t$ , the sequence $G_n$ grows geometrically, leading to a contradiction where the value of the function at a specific negative point must be infinite, which is impossible for a real-valued function.
Explicit Construction for $t=1/3$ : To apply the general criterion to the specific case $t=1/3$ , the author explicitly constructs the required parameters:
- $C = 8/3$
- $\eta = 10/9$
- Three pairs of points: $(1/9, 4/9)$ , $(0, 2/3)$ , and $(0, 1)$ .
- Coefficients: $\mu_1 = 3$ , $\mu_2 = 2$ , $\mu_3 = 4/3$ .
  The author verifies that the linear combination of $J_{1/3, \phi}$ terms using these parameters yields the required identity, thereby satisfying the conditions of Proposition 2.1.

Key Contributions and Results

Theorem 1.3: The primary result is the negative answer to Páles' question. The paper proves that for $t = 1/3$ , every non-symmetric $t$ -convex function $f: \mathbb{R} \to \mathbb{R}$ is necessarily $t$ -convex.
Generalization: The methodology establishes that the set $T$ of all $t \in (0, 1)$ for which non-symmetric $t$ -convexity implies $t$ -convexity contains $1/2$ (trivial) and $1/3$ (proven).
Structural Properties of $T$ :
- Symmetry: The set $T$ is symmetric around $1/2$ (i.e., $t \in T \iff 1-t \in T$ ).
- Density: The set $T$ is dense in the interval $(0, 1)$ . This is proven using a construction involving finite words and polynomial mappings that generate a dense set of parameters satisfying the convexity condition.
- Specific Elements: The paper notes that similar computations show $1/k \in T$ for $k \in \{4, 5, 6\}$ , and specific algebraic numbers like $\frac{\sqrt{5}-1}{2}$ and $\frac{3-\sqrt{5}}{2}$ belong to $T$ .

Significance and Open Questions
The paper resolves a specific open problem posed by Zsolt Páles regarding the case $t=1/3$ , confirming that the failure of the implication for transcendental $t$ does not extend to this rational case. The work provides a constructive method (Proposition 2.1) to determine if other rational or algebraic values of $t$ belong to the set $T$ .

The author concludes by noting that while $T$ is dense and contains many algebraic numbers, it remains an open question whether $T$ contains all algebraic numbers in $(0, 1)$ . The paper does not propose further applications or experimental directions, focusing strictly on the theoretical characterization of these functional inequalities.

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