A contribution to the Aleksandrov conservative distance problem in two dimensions

Abstract

Abstract. Let E be a two-dimensional real normed space. In this paper we show that if the unit circle of E does not contain any line segment such that the distance between its endpoints is greater than 1, then every transformation φ : E → E which preserves the unit distance is automatically an affine isometry. In particular, this condition is satisfied when the norm is strictly convex.

Research on coal and gangue segmentation based on MFCCM‐Mask R‐CNN

“…The evaluation metrics of other current segmentation network models on this paper's data set are demonstrated in Table 7, and this paper's method achieves optimal results in segmentation recall and F1 metrics, reaching 99.49% and 98.42%, respectively. Compared to other segmentation networks yoloact, 47 Unet 48 and Deeplab V3+, 49 the accuracy of segmentation is 1.08%, 0.76%, and 0.91% higher, respectively. Although yolov7 50 has high recognition accuracy, the pixel information in the generated box frame is not well utilized.…”

Mappings of Conservative Distances In -Normed spaces ()

“…(see [7,8]; see also Figure 1). It is known [14] that a real normed linear space X is strictly convex if and only if any two-dimensional subspace of X has the following property.…”

“…As far as we know, this problem is still far from being solved. It was solved only for a few concrete two-dimensional normed spaces (see [7] concerning strictly convex normed spaces and [10] for a nonstrictly convex normed space). For modified versions of the Aleksandrov question, there are two known results.…”

“…7 is also an r-probe and the four points f (x 0 ), f (y 1 ), f (y 2 ), f (y 3 ) have distance r from each other. Two of the points are on the same side of the line passing through the other two points.…”