On the set of continuously differentiable concave extensions of a Boolean function
Vestnik rossijskih universitetov. Matematika, Tome 30 (2025) no. 149, pp. 5-14
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This paper is devoted to the study of the existence of extremal elements of the set of continuously differentiable concave extensions to the set $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,\ldots,x_n)$, as well as finding the cardinality of the set of continuously differentiable concave extensions to $[0,1]^n$ of the Boolean function $f_{B}(x_1,\ldots,x_n).$ As a result of the study, it is proved that, firstly, for any Boolean function $f_{B}(x_1,\ldots,x_n)$ among its continuously differentiable concave extensions to $[0,1]^n$ there is no maximal element, secondly, if the Boolean function $f_{B}(x_1,\ldots,x_n)$ has more than one essential variable, then among its continuously differentiable concave extensions to $[0,1]^n$ there is no minimal element, and if the Boolean function is constant or has only one essential variable, then among its continuously differentiable concave extensions to $[0,1]^n$ there is a unique minimal element, the explicit form of which is given in the paper. It was also established that the cardinality of the set of continuously differentiable concave extensions to $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,\ldots,x_n)$ is equal to the continuum.
Keywords:
continuously differentiable concave extension of a Boolean function, extremal elements of a set, cardinality of a set
@article{VTAMU_2025_30_149_a0,
author = {D. N. Barotov and R. N. Barotov},
title = {On the set of continuously differentiable concave extensions of a {Boolean} function},
journal = {Vestnik rossijskih universitetov. Matematika},
pages = {5--14},
publisher = {mathdoc},
volume = {30},
number = {149},
year = {2025},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VTAMU_2025_30_149_a0/}
}
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%0 Journal Article %A D. N. Barotov %A R. N. Barotov %T On the set of continuously differentiable concave extensions of a Boolean function %J Vestnik rossijskih universitetov. Matematika %D 2025 %P 5-14 %V 30 %N 149 %I mathdoc %U http://geodesic.mathdoc.fr/item/VTAMU_2025_30_149_a0/ %G ru %F VTAMU_2025_30_149_a0
D. N. Barotov; R. N. Barotov. On the set of continuously differentiable concave extensions of a Boolean function. Vestnik rossijskih universitetov. Matematika, Tome 30 (2025) no. 149, pp. 5-14. http://geodesic.mathdoc.fr/item/VTAMU_2025_30_149_a0/