{'fraction_non_alphanumeric': 0.026923076923076925, 'fraction_numerical': 0.002307692307692308, 'mean_word_length': 5.023148148148148, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Directional-Quasi-Convexity, Asymmetric Schur-Convexity and Optimality of Consecutive Partitions

The current paper has the following distinct goals: 
 
1. To generalize standard quasi-convexity to obtain a weaker property of functions that suffices for optimality of extreme points. Specifically, it is required that intersections of the level sets of the function with line segments having direction in a prescribed set are convex. 
 
2. To characterize Schur-convexity for symmetric functions through the above generalization of quasi-convexity, thereby obtaining a previously unknown relationship of Schur-convexity and standard convexity. The new characterization is used to extend the definition of Schur-convexity to functions which are not symmetric. 
 
3. To obtain new sufficient conditions for the optimality of consecutive partitions, by using the new definition of Schur-convexity for functions that are not necessarily symmetric. 
 
The conclusion that Schur-convexity is an instance of a convexity property that implies the optimality of extreme points, unifies two approaches that were used in the literature to prove optimality of subsets of the domain of real-valued functions. The two tools---quasi-convexity and Schur-convexity---were previously considered as distinct techniques.