I will provide a proof of a theorem by Minkowski, which, to my knowledge, is simpler than the previously known proofs. The theorem states:

If a convex region B, symmetric with respect to the origin O, in the plane of the rectangular coordinate system, has an area greater than 4, then B contains a lattice point other than O as an interior point.

About half a century ago, Minkowski's work "Geometry of Numbers" was published, in which he raised, among other things, a question that has been much discussed ever since. We intend to answer this question in the present work; it is related to the following theorem by Minkowski, proven in the aforementioned work:

If $\xi_{1},...,\xi_{n}$ are homogeneous linear forms in the variables $x_{1},..., x_{n}$, with real coefficients and determinant 1, then there exist integers $x_{1},..., x_{n}$ that do not all vanish and for which
          $|\xi_{1}| ≤ 1 , ..., |\xi_{n}| ≤ 1$.

Several authors have addressed this conjecture, but none have been able to provide a definitive answer. H Jansen (1909) proved its validity for n ≤ 6, Th Schmidt (1933) for n ≤ 7, and O-H Keller (1937) for n ≤ 8. However, regarding these works, see the critical remarks of O Perron (1940), who gave a clear proof for n ≤ 9 (1941). B Levi (1911), S L Van Oss (1915), and C L Siegel (1922) have dealt with the general conjecture, but their proofs contain gaps or are not fully developed. J F Koksma (1936) and L J Mordell (1936) gave reports on the state of the entire problem. In the present work, the validity of the conjecture is proven for every n.

The scientific work of Hajós culminated in his famous proof (1941) of Minkowski's conjecture (1896) concerning the lattice covering of the n-dimensional space with cubes. The proof rests on his theorem of the factorisation of Abelian groups the importance of which was later recognised in the international mathematical literature. Rédei called it the second fundamental theorem of the theory of Abelian groups.

After discussing the necessity, advantages, and principles of error estimation, the aim is to systematically present the well-known elements of error estimation and to supplement them in some places. Applications of probability theory are not touched upon. In the sense of approximation mathematics, only practical purposes are served.

The author considers various problems connected with the factorisation of abelian groups and the filling of space by congruent hypercubes. Let A and B be two subsets of a given group G; then we write AB = G if corresponding to any $g \in G$ there exist unique elements $a \in A, b \in B$, such that $ab = g$, and we say that G is divisible by A and by B. A subset S consisting of elements $1, a, a^{2},…,a^{k−1}$ is called a series; when k is the order of a, S is a cyclic subgroup and is then said to be periodic. More generally, a subset A is called periodic if $aA = A$ for some $a \in G$ and A is then divisible by the cyclic subgroup generated by a. The Minkowski conjecture concerning homogeneous linear forms is equivalent to the author's theorem [1941] that if a finite abelian group is factorisable as $G = S_{1}S_{2} ... S_{n}$ where $S_{1}, S_{2}, …, S_{n}$ are series, then at least one $S_{i}$ is periodic. Various extensions of this result are discussed ...

His name is associated with the proof of a famous conjecture by H Minkowski, which resisted the attempts of the greatest for many years; his theorem is cited everywhere as the Minkowski-Hajós theorem. In connection with the proof of the Minkowski conjecture, he expanded the classical theory of finite abelian groups with a breakthrough discovery. With this work he gained international fame. He achieved valuable results in the fields of discrete geometry, lattice point geometry, Bolyai-Lobachevsky geometry, the theory of geometric constructions, graph theory, nomography, and numerical analysis.

Hajós was also an excellent teacher, his lectures had a country-wide fame. He was able to present the matter very clearly, very interesting and, at the same time, with highest logical precision. His book, "Introduction to Geometry" (1960) was published four times in Hungarian and translated into German (1970).

Hajós loved his country profoundly. After many visits at foreign universities and congresses, he enjoyed to return to his pupils and tried to enrich and to improve the Hungarian mathematical life. He left a great lack behind him and we shall always cherish his memory.