Hlawka's Inequality

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常用不等式（匡继昌第三版）第2页
5. Hlawka不等式
$$\left|a_{1}\right|+\left|a_{2}\right|+\left|a_{3}\right|-\left|a_{1}+a_{2}\right|-\left|a_{2}+a_{3}\right|-\left|a_{3}+a_{1}\right|+\left|a_{1}+a_{2}+a_{3}\right| \geqslant 0\tag{1.7}$$
当 $a_1, a_2, a_3$ 为 $R^m$ 中的向量或实赋范线性空间中的向量, (1.7) 式仍成立. 见 [305] 1965,72:753-754.
2000 年 Takahasi,S.E. 等推广了 (1.7) 式并证明了在 Banach 空间 $(X,\|\cdot\|)$ 中 (1.7) 式与下述 Djoković 不等式等价:
\[
\sum_{1 \leqslant i_1<\cdots<i_k \leqslant n}\left\|\sum_{m=1}^k a_{i_m}\right\| \leqslant\left(\begin{array}{l}
n-2 \\
k-1
\end{array}\right) \sum_{k=1}^n\left\|a_k\right\|+\left(\begin{array}{l}
n-2 \\
k-2
\end{array}\right)\left\|\sum_{k=1}^n a_k\right\|,(2 \leqslant k \leqslant n-1),
\]
见 [303] 2000,3(1): 63-67 和 [398] 2000,1(3): 343-350.
1963 年 Freudenthal, H 提出: 设 $a_k \in R^m$. 对于什么样的 $n$, 成立
\[
\sum_{k=1}^n\left|a_k\right|-\sum_{1\leqslant i<j\leqslant n}\left|a_i+a_j\right|+\sum_{1 \leqslant i<j<k \leqslant n}\left| a_i+a_j+a_k\right|-\cdots+(-1)^{n-1}\left| \sum_{k=1}^n a_k \right|\geqslant 0
\]
1997 年 Jiang-cheng 证明上式仅对 $n=1, n=2$ (Minkowski 不等式) 和 $n=3$ (即 (1.7) 式) 成立. 见 Vietnam J. Math, 1997,25(3):271-273.
1964 年 Adamovic 将 (1.7) 式推广为
\[\tag{1.8}
\sum_{1\leqslant i<j\leqslant n}\left|a_i+a_j\right| \leqslant(n-2) \sum_{k=1}^n\left|a_k\right|+\left|\sum_{k=1}^n a_k\right| \text {. }
\]
式中 $a_k \in R^m$, 见 [355] 1964,1(16): 39-43.

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mathworld
Let $V$ be an inner product space and let $x,y,z$ in $V$. Hlawka's inequality states that
\[\|x+y\|+\|y+z\|+\|z+x\|\le\|x\|+\|y\|+\|z\|+\|x+y+z\|,\]
where the norm $\|z\|$ denotes the norm induced by the inner product.

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Hlawka’s functional inequality
This article is published with open access at Springerlink.com
The paper is devoted to the functional inequality (called by us Hlawka’s functional inequality)
$$f(x+y)+f(y+z)+f(x+z)≤f(x+y+z)+f(x)+f(y)+f(z)$$
for the unknown mapping f defined on an Abelian group, on a linear space or on the real line. The study of the foregoing inequality is motivated by Hlawka’s inequality:
$$\|x+y\|+\|y+z\|+\|x+z\|\leq\|x+y+z\|+\|x\|+\|y\|+\|z\|,$$
which in particular holds true for all $x, y, z$ from a real or complex inner product space.

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Hlawka's inequality proof in with integrals
Proof of Hlawka's inequality for complex numbers.
cut-the-knot — Hlawka's Inequality

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Djoković 1963

GENERALIZATIONS OF HLAWKA'S INEQUALITY
Dmgomir Ž. Đoković, Beograd

1. Introduction

Let $E$ be complex pre-Hilbert (unitary) space. We shall denote the norm of $a \in E$ by $|a|$. The following identity is due to E. Hlawka [1]:
\[
\begin{aligned}
&\left(\left|a_1\right|+\left|a_2\right|+\left|a_3\right|-\left|a_2+a_3\right|-\left|a_3+a_1\right|-\left|a_1+a_2\right|+\right.\\
+&\left.\left|a_1+a_2+a_3\right|\right) \times\left(\left|a_1\right|+\left|a_2\right|+\left|a_3\right|+\left|a_1+a_2+a_3\right|\right)=\\
=&\left(\left|a_2\right|+\left|a_3\right|-\left|a_2+a_3\right|\right)\left(\left|a_1\right|-\left|a_2+a_3\right|+\left|a_1+a_2+a_3\right|\right)+\\
+&\left(\left|a_3\right|+\left|a_1\right|-\left|a_3+a_1\right|\right)\left(\left|a_2\right|-\left|a_3+a_1\right|+\left|a_1+a_2+a_3\right|\right)+\\
+&\left(\left|a_1\right|+\left|a_2\right|-\left|a_1+a_2\right|\right)\left(a_3|-| a_1+a_2|+| a_1+a_2+a_3|\right)
\end{aligned}
\]
where $a_1, a_2, a_3$ are arbitrary elements of $E$. From this it follows that
\[
\left|a_1\right|+\left|a_2\right|+\left|a_3\right| -\left|a_2+a_3\right|-\left|a_3+a_1\right|-\left|a_1+a_2\right|+\left|a_1+a_2+a_3\right| \geq 0 .
\]
This is Hlawka's inequality.