The main focus is on **tensor valuations** which are tensor-valued additive functionals on the families of convex or more general sets. Special cases are the classical intrinsic volumes (volume, surface area, length, Euler number) which are **Minkowski tensors** of rank zero.

In stochastic geometry, the focus has in recent years turned to Minkowski tensors of rank one or higher which provide information about position, orientation and shape of spatial structures.

**WP1.1:** Tensor valuations and integral geometry**WP1.2:** Isoperimetric inequalities with tensor constraints**WP1.3:** Uniqueness of measurement functions in integral formulae**WP1.4:** Stereology of tensors

In the second funding period of CSGB, we take advantage of very recent theoretical advances. These concern the algebraic structure (product, convolution, Fourier transform) of tensor valuations and the locally defined Minkowski tensors.

- Further and stronger results for
**integral geometry**of Minkowski tensors are expected, using the new algebraic results. One of the key ideas is to use**trace-free tensors**which are closely related to spherical harmonics. Indeed, the mathematical theory suggests that trace-free tensors are in several respects more natural than the ordinary tensors. - The new locally defined Minkowski tensors will be used in an attempt to establish a
**principal rotational formula**for Minkowski tensors. Such a formula will give a class of measurement techniques in optical microscopy a common formulation and special cases of the formula are expected to provide new microscopy techniques.

In the second funding period, it is also planned to transfer local stereological estimators of Minkowski tensors to particle populations. Important statistical issues are here

- the development of
**confidence regions**for shape and orientation, based on finite samples of particles - statistical comparison of shape and orientation in two
**particle populations** **asymptotics**in an expanding window regime

- Hasselholt, S., Hahn, U., Jensen, E.B.V. & Nyengaard, J.R. (2019): Practical implementation of the planar and spatial rotator in a complex tissue – the brain.
*J. Microsc.***273**, 26–35. - Kiderlen, M. & Jensen, E.B.V. (2017, eds.):
*Tensor Valuations and their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics***2177.**Springer. - Kousholt, A., Ziegel, J.F., Kiderlen, M. & Jensen, E.B.V. (2017): Stereological estimation of mean particle volume in R^3 from vertical sections. In
*Tensor Valuations and their Applications in Stochastic Geometry and Imaging*(eds. M. Kiderlen and E.B.V. Jensen), Lecture Notes in Mathematics**2177**, Springer, pp. 423-434. - Larsen, N.Y., Ziegel, J.F., Nyengaard, J.R. & Jensen, E.B.V. (2019): Stereological estimation of particle shape from vertical sections.
*J. Microsc.***275**, 183–194. - Svane, A.M. & Jensen, E.B.V. (2017): Rotational Crofton formulae for Minkowski tensors and some affine counterparts.
*Adv. Appl. Math.***91**, 44-75.