Extremal problems for non-periodic splines on real domain and their derivatives

We consider the Bojanov-Naidenov problem over the set $\sigma_{h,r}$ of all non-periodic splines $s$ of order $r$ and minimal defect with knots at the points $kh$, $k \in \mathbb{Z}$. More exactly, for given $n, r \in \mathbb{N}$; $p, A > 0$ and any fixed interval $[a, b] \subset \mathbb{R}$ we solve the following extremal problem $\int\limits_a^b |x(t)|^q dt \rightarrow \sup$, $q \geqslant p$, over the classes $\sigma_{h,r}^p(A) := \bigl\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, \| s \|_{p, \delta} \leqslant A \| \varphi_{\lambda, r} \|_{p, \delta}, \delta \in (0, h], \tau \in \mathbb{R} \bigr\}$, where $\| x \|_{p, \delta} := \sup \bigl\{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a \leqslant \delta \bigr\}$, and $\varphi_{\lambda, r}$ is $(2\pi / \lambda)$-periodic spline of Euler of order $r$. In particularly, for $k = 1, ..., r - 1$ we solve the extremal problem $\int\limits_a^b |x^{(k)}(t)|^q dt \rightarrow \sup$, $q \geqslant 1$, over the classes $\sigma_{h,r}^p (A)$. Note that the problems (1) and (2) were solved earlier on the classes $\sigma_{h,r}(A, p) := \bigl\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, L(s)_p \leqslant AL(\varphi_{n,r})_p, \tau \in \mathbb{R} \bigr\}$, where $L(x)_p := \sup \bigl\{ \| x \|_{L_p[a, b]} \colon a, b \in \mathbb{R}, |x(t)| > 0, t \in (a, b) \bigr\}$. We prove that the classes $\sigma_{h,r}^p (A)$ are wider than the classes $\sigma_{h,r}(A,p)$. Similarly we solve the analog of Erdös problem about the characterisation of the spline $s \in \sigma_{h,r}^p(A)$ that has maximal arc length over fixed interval $[a, b] \subset \mathbb{R}$.

Isosceles Sets

In 1946, Paul Erdős posed a problem of determining the largest possible cardinality of an isosceles set, i.e., a set of points in plane or in space, any three of which form an isosceles triangle. Such a question can be asked for any metric space, and an upper bound ${n+2\choose 2}$ for the Euclidean space ${\Bbb E}^{n}$ was found by Blokhuis. This upper bound is known to be sharp for $n=1$, 2, 6, and 8. We will consider Erdős' question for the binary Hamming space $H_{n}$ and obtain the following upper bounds on the cardinality of an isosceles subset $S$ of $H_{n}$: if there are at most two distinct nonzero distances between points of $S$, then $|S|\leq{n+1\choose 2}+1$; if, furthermore, $n\geq 4$, $n\ne 6$, and, as a set of vertices of the $n$-cube, $S$ is contained in a hyperplane, then $|S|\leq{n\choose 2}$; if there are more than two distinct nonzero distances between points of $S$, then $|S|\leq{n\choose 2}+1$. The first bound is sharp if and only if $n=2$ or $n=5$; the other two bounds are sharp for all relevant values of $n$, except the third bound for $n=6$, when the sharp upper bound is 12. We also give the exact answer to the Erdős problem for ${\Bbb E}^{n}$ with $n\leq 7$ and describe all isosceles sets of the largest cardinality in these dimensions.