**6. Discussion**

Using the Barber's extremal principle, we derived explicit analytical relations for all essential contact properties of an indenter of "arbitrary" shape (under restriction that it should be close to an axisymmetric one). The solution provides the dependency of the normal force on the indentation depth, the size and the shape of the contact area, and the pressure distribution in the contact area. The derivation has been carried out under two assumptions (which are the main sources of deviation from the exact solution): (1) Fabrikant's approximation for the stress of a flat-ended punch with arbitrary cross-section and (2) assumption of a small deviation of the indenter shape from an axial one. However, an acceptable accuracy is obtained even with relatively large deviations from axial symmetry, e.g., the eccentricity of the contact area for a paraboloid is given accurately up to an eccentricity of approximately 0.7. The deviation of normal force in a contact of a pyramid indenter with square cross-section is of about 4.6%. The whole calculation resembles the Method of Dimensionality Reduction very much and can be considered as its generalization for non-axially symmetric contacts. A central approximation used in this paper is Fabrikant's approximation for the pressure under a flat punch. This approximation could be further improved by using higher-order corrections obtained by Golikova and Mossakovskii for the pressure distribution under plane stamps of nearly circular cross-section [15].

**Funding:** This work has been conducted under partial financial support from the German Research Society (DFG), Project PO 810/66-1. It was partially supported by the Tomsk State University Development Programme («Priority-2030»).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The author thanks E. Willert for discussions and Q. Li for providing numerical data for indentation of a pyramid. We acknowledge support by the German Research Foundation and the Open Access Publication Fund of TU Berlin.

**Conflicts of Interest:** The author declares no conflict of interest.
